Midterm 2 Study

Question 1

What is a Sub Space? List the 3 conditions.

Answer 1

A subset S of vectors in R^n, is also a subspace of R^n if these 3 conditions hold:
1 - The zero vector exists in S
2 - For any two vectors in S, the sum of those two vectors also exists in S. (closure under addition). i.e. vec1, vec2 exist in S, (vec1 + vec2) exists in S.
3 - For any vector in S, and any real number scalar in S, the product of them exists in S. (closure under scalar multiple) i.e. vec1 exists in S, r is a real number, r*vec1 exists in S.

Question 2

What is an Elementary Matrix? Are they invertible?

Answer 2

An Elementary Matrix is the matrix that results from performing exactly ONE row operation (add, mult, swap) on an Identity Matrix.

-If I perform a row operation R, on identity matrix In, I get an elementary matrix E. If I then multiply any matrix A, by E, (E*A) then that is the same as performing row operation R on A.

E must be on LEFT SIDE of multiplication for row operation equivalent.

• Elementary Matrices are like encoded row operations.
• Every elementary Matrix is invertible.

Question 3

How are span and subspace related?

Answer 3

If a set of vectors exists in R^n, then the span of those vectors is a subspace of R^n.

Question 4

What is span?

Answer 4

Given a set of vectors {vec1, vec2……vecm} that exist in R^n, the span is the set of all possible linear combinations of those vectors, denoted span{vec1, vec2…..vecm}.

If span{vec1, vec2,….vecm}=R^n, then we can say that set spans R^n. This means that all the linear combinations of that set of vectors can recreate the entire dimensional space of R^n

Question 5

What is column space?

Answer 5

Given a matrix A, the column space of A, denoted col(A) is the vector space created by all possible linear combinations of the column vectors of A.

if A=[vec1, vec2…vecn], then col(A)=span[vec1, vec2…vecn].

Question 6

What is the difference between span and column space?

Answer 6

Span specifically refers to the dimensional space created by all possible linear combinations OF A SET OF VECTORS, which could be either column of row vectors.

Column space specifically refers to the dimensional space created by all possible linear combinations of the column vectors OF A MATRIX.

They are very similar, except span takes a set of vectors as input, and column space (col) takes a matrix as input.
They both output a dimensional space.

Question 7

How can you check if a vector is in the column space of a matrix?

Answer 7

Given a matrix A, and vector Bvec, if there is a solution to AXvec=Bvec, then Bvec exists in col(A). First, make sure the vector and matrix are the same dimension (or else you can’t augment them). Then augment the matrix A with Bvec (add it as a new column to the rightmost side in A). Reduce to echelon form. If there is any solution, then Bvec exists in col(A). If there is no solution it does not.

Question 8

What is a standard basis?

Answer 8

The standard basis for R^n, is the set of vectors that when put into matrix form, create the identity matrix for R^n. The vectors are denoted by an ‘e’

i.e. evec1 has a 1 in the first component (rest all zeros), evec2 has a 1 in the second component…evecn has a 1 in the nth component.

Question 9

What is the basis of a subspace?

Answer 9

Given a subset S of R^n, that is also a subspace of R^n, there is a subset of vectors B that that is a basis for S if:
1 - S=span(B)
2 - B is linearly independent.

This is saying that there is always a finite set of linearly Independent vectors that can reconstruct the subspace S. Any vector in S can be written as a UNIQUE linear combination of the basis vectors.

FACT: Every non-zero subspace has a basis.

Question 10

Given a matrix A, how do you find the basis of col(A)?

Answer 10

Have matrix B be an echelon form of A. Notice where the pivot columns are in B. The set of column vectors in A that correspond to the position of the pivot columns in B is the basis of col(A).

Question 11

What is the dimension of a basis?

Answer 11

The number of vectors in that basis is the dimension.

i.e. if a basis contains 3 vectors, then it’s dimension is R^3.

Question 12

Given a matrix A, how do you find the basis of null(A)?

Answer 12

Let matrix B, be a RREF of matrix A. Find the solution to A(xvec)=0vec. (homogeneous equation). The set of vectors that satisfies the homogeneous equation is the basis of null(A).

The number of vectors in that set will be equal to the number of free variables in the RREF.

Question 13

What is the Basis Theorem?

Answer 13

Let S be a subset of R^n. Let S also be a non-zero subspace that exists in R^m (so m<=n). Let B, be a set of vectors in S, from 1 to m. Then if B is linearly Independent, it is a basis for S. Similarly, if B spans S (span{B}=S), then B is a basis for S.

This makes sense because if you have number of vectors in a set B equal to the number of dimensions of the subspace S that B exists in, and B is linearly independent, then none of the vectors are linear combinations of each other. This means each vector represents a unique dimension, and if there are m vectors, there will me m dimensions in it’s span. It is a basis because it is the minimal set needed to recreate m dimensions (there are no redundant (linearly DEPendent) vectors).

If B spans S, B is a basis of S because B has exactly m vectors in it, and it is spanning a subspace of dimension m, so B cannot have any redundant (linearly DEPendent) vectors in it, because that would mean the numbers of vectors in it would have to be more than m. If B had less than m vectors in it, than it would be impossible to span dimension m. (To span dimension m, you need at least m vectors.)

If B has more than m vectors in it, it is not a basis.
If B has less than m vectors in it, it cannot span S.

Question 14

What is the definition of an Invertible Matrix?

Answer 14

Given an nn matrix A, if there exists an nn matrix C such that AC=CA=Identity, then C is an inverse for A. We say that A is invertible.

Question 15

How do you find the inverse of a Matrix, if it has one?

Answer 15

Given an nn matrix A, augment it on the right with the nn Identity matrix. Reduce this augmented matrix until the left half (The A matrix) is in Reduce Row Echelon Form. The right half will be the inverse of A.

In the process of row reducing, if there is ever a row of zeros on in the left half, you can stop, and conclude that matrix A does not have an inverse. (This is because that row of zeros will never go away, and thus never be in RREF).

Question 16

What is the Spanning Theorem?

Answer 16

Given a m*n matrix A, the following are equivalent (If one of these is true, they’re all true. If one is false, they’re all false).
1 - A(xvec)=bvec has a solution for all bvec that exist in R^m.
2 - There is a linear combination of the column vectors in A that can equal any bvec in R^m.
3 - The columns of A span R^m
4 - The echelon form of A has a pivot in every row.
5 - The linear transformation T:R^n–>R^m with standard matrix A is onto.

Question 17

What is the Linear Independence Theorem?

Answer 17

Given a k*n matrix A, the following are equivalent (If one of these is true, they’re all true. If one is false, they’re all false).
1 - A(xvec)=0vec has only the trivial solution.
2 - The columns of A are linearly Independent.
3 - Let B be an echelon from of A, then B has a pivot in each column.
4 - Let T: R^m. –>R^n be the linear transformation whose standard matrix is A. Then T is one-to-one.

Question 18

What is the definition of dim?

Answer 18

Given a subspace S that exists in R^n, dim(S)=n.

The dim() function takes a dimensional space as input, and outputs it’s number of dimensions.

i.e. If subspace S2 exists in R^3, then dim(S2)=3.

Question 19

What is the definition of Rank?

Answer 19

Given a k*n matrix A, then rank(A) is defined as rank(A)=dim(col(A)).

Rank is a function that takes a matrix as input, and outputs the number of dimensions of that Matrix’s column space.

i.e. if col(A)=R^5, then rank(A)=5.

Question 20

What is the Rank-Nullity Theorem?

Answer 20

Let matrix A be k*n. Then Rank(A)+dim(Null(A))=n.

Since Rank(A)=the number of pivot columns in an echelon form of A, and dim(Null(A))= the number of non-pivot columns in an echelon form of A, Rank(A)+dim(Null(A))=the total number of columns in A.

Question 21

What is the definition of Transpose?

Answer 21

Given matrix A=[a(ij)] be mn. Then the transpose of A, denoted A^T=[a(ji)], is the nm matrix whose columns are the rows of A. That is A^T=[a(ji)], 1<=j<=n, 1<=i<=m.

i.e. Flip matrix A by 180 degrees on Z axis (so top row is now bottom row, and bottom row is top row), then rotate clockwise by 90 degrees on XY axis. (So the original top row is now the first column).

Question 22

What is a transformation?

Answer 22

A transformation (or function, or mapping) T, from R^n to R^m, is a rule that assigns to each vector xvec in R^n, exactly one vector T(xvec) from R^m.

A transformation is a function that takes all vectors in a dimensional space, inputs them into some expression, and outputs vectors in the same or different dimensional space.

i.e. R^3–>R^2 T(xvec)=A(xvec). This transformation will take a vector xvec in R^3 as input, and multiply a matrix A in R^2 with it. The result will be a vector in R^2.

Question 23

What is the domain?

Answer 23

In a transformation T: R^n–>R^m, the set of all vectors in R^n is called the domain of T.

The domain of T is all possible input vectors for T.

Question 24

What is the codomain?

Answer 24

In a transformation T: R^n–>R^m, the set of all vectors in R^m is called the codomain of T.

The codomain of T is all possible output vectors for T.

Question 25

What is image?

Answer 25

In a transformation T: R^n–>R^m, the vector T(xvec) is called the image of xvec under T.

Given an input vector xvec, for a transformation, the resulting output vector T(xvec) is called the image of xvec.

Question 26

What is range?

Answer 26

In a transformation T: R^n–>R^m, the set of all vectors T(xvec) is called the range of T.

The set of all images for a transformation T, is called the range of T.

The set of resulting output vectors for a transformation T is called the range of T.

Question 27

What is a Linear Transformation?

Answer 27

A transformation T: R^n–>R^m, is called a Linear Transformation if:
1 - T(uvec+vvec)=T(uvec)+T(vvec), for all uvec, vvec exist in R^n.
2 - T(cuvec)=cT(uvec), for all uvec exist in R^n, and all exist in R.

Question 28

What is a standard matrix?

Answer 28

Given a linear transformation T: R^n–>R^m, the standard matrix A, for T is a unique, mn matrix such that T(xvec)=Axvec for all xvec exists in R^n.

This means that the columns of A are made up of the transformation of the elementary vectors from 1 to n
A=[T(elemvec1), T(elemvec2),…T(elemvecn)]

Each column in matrix A is the linear transformation of the corresponding column vector in the identity matrix from 1 to n

Question 29

What is the definition of Row Space?

Answer 29

Given a k*n matrix A, the row space of A, denoted row(A), is the span of the rows of A, when considered as column vectors.

Question 30

What is the Existence-Uniqueness Theorem?

Answer 30

A Linear System is consistent if and only if an echelon form of the augmented matrix has NO row in the form [0,0…b], where b is a real number.

This makes sense because it is impossible for the sum of zeros to equal a non-zero number.

Question 31

What is the Linear Independence Theorem?

Answer 31

Let A be an m*n matrix. Then the following are equivalent: (If one of these is true, they’re all true. If one is false, they’re all false).

1 - The matrix equation A(xvec)=zerovec has only the trivial solution.

2 - The columns of A are linearly independent.

3 - The echelon form of A has a pivot in every column.

4 - Let T:R^n–>R^m be the linear transformation with standard matrix A. Then T is one-to-one.

Question 32

What is the Invertible Matrix Theorem?

Answer 32

Let A be an nn (NN, must be square!!!!) matrix. Then the following are equivalent. (If one of these is true, they’re all true. If one is false, they’re all false).
1 - A is invertible (i.e. A^-1 exists)
2 - The RREF of A is the Identity Matrix.
3 - The echelon form of A has n pivot positions.
4 - The homogeneous equation Ax=0 has only the trivial solution.
5 - The columns of A are linearly independent.
6 - The linear transformation T:R^n–>R^n with standard matrix A is one to one.
7 - The equation A(xvec)=bvec has exactly one solution for each bvec exists in R^n.
8 - The columns of A span R^n.
9 - The linear transformation T:R^n–>R^n with standard matrix A is onto.
10 - There is an nn matrix C such that AC=Identity_n.
11 - There is an nn matrix D such that DA=Identity_n.
12 - A^T is invertible.
13 - The columns of A form a basis for R^n.
14 - Col(A) = R^n.
15 - Rank(A) = n
16 - dim(Null(A))=0.
17 - Null(A) = {zerovec}
18 - dim(Row(A))=n
19 - Row(A) = R^n

Question 33

What are the properties of Matrix Inverses?

Answer 33

1 - (A^-1)^-1=A
2 - In^-1=In
3 - (AB)^-1=B^-1A^-1 (swapped!)
4- (A^-1)^T=(A^T)^-1

Question 34

What is a Dimension?

Answer 34

Dimension is the number of components in a vector. (Rows of a matrix).

i.e. If vec1={v1,v2,v3,v4}, then the dimension of that vector is 4.

Question 35

What are the implications of two subsets S1, S2 of R^n, that are subspaces (not necessarily of exactly R^n), with S1 being a subset of S2?

Answer 35

This is corollary 34.

Given two subsets S1, S2 of R^n, that are also subspaces (not necessarily of exactly R^), with S1 being a subset of S2. Then dim(S1)<=dim(S2), and dim(S1)=dim(S2), if and only if S1=S2.

Question 36

Given a subset S of R^n, that is a subspace of dimension m, and a set of vectors in S, B={vec1…veck}, what are the implications of this?

Answer 36

This is Corollary 35

1 - If k

Question 37

What is a canonical basis?

Answer 37

Give an m*n matrix A, the rows of A^T, are the columns of A, therefore Col(A)=Row(A^T). The non-zero rows of the RREF of A^T form the canonical basis of A.

Question 38

What is the formula for the inverse of a 2x2 matrix? What is a fast way to check if it is invertible?

Answer 38

Given matrix A=«a>|<b>>, and given that A is invertible, then the formula is:
A^-1 = (1/(ad-bc))*<|></b>

If ad-bc is not equal to 0, then it is invertible. </b></a>

Question 39

Given matrix A, how do you easily find the solution to Ax=b using the inverse of A and no row reduction?

Answer 39

If given the inverse of A, and some vector b, then we can easily solve for x by simply rearranging the general solution to get x=(A^-1)b.

We do not need to augment A with b, so no row reduction!

Question 40

How can the general solution Ax=B, be rewritten to include the A^-1?

Answer 40

Given invertible matrix A, x=(A^-1)b is the only solution to Ax=b.

Question 41

What is a Null Space?

Answer 41

Let A be an mn matrix. Then the null space is A is the set of all solutions to the homogeneous equation. Axvec=0vec.

The set of all vectors that when multiplied by A, equal the zero vector.