Vocabulary Flashcards
m by n matrix
Size of the matrix which indicates the number of rows and columns. If m and n are positive integers, an m x n matrix is a rectangular array of numbers with m rows and n columns.
row equivalent matrices
Two matrices with the same solution set.
In other words, one matrix can be converted to the other using elementary row operations.
echelon form of a matrix
A rectangular matrix is in echelon form (or row echelon form) if it has the following three properties:
1. All nonzero rows are above any rows of all zeroes.
2. Each leading entry of a row is in a column to the right of the leading entry of the row above it.
3. All entries in a column below a leading entry are zeroes.
reduced row echelon form (RREF) of a matrix
A matrix in echelon form which satisfies the following additional conditions:
1. The leading entry in each nonzero row is 1.
2. Each leading 1 is the only nonzero entry in its column.
scalar
A (real) number used to multiply either a vector or a matrix.
zero vector
The unique vector, denoted by 0, such that u+0=u for all u. In Rn, 0 is a vector whose entries are all zero.
Ex: v0 = (0,0,0)
linear combination of a set of vectors
A sum of scalar multiples of vectors. The scalars are called weights. Given vectors v1, v2, …, vp in Rn and given scalars c1, c2, …, cp, the vector y defined by y=c1v1 + … + cpvp is a lin. comb. of v vectors with c weights.
pivot position, pivot column
pivot position - In a matrix A it is a location in A that corresponds to a leading # other than zero, or in the reduced echelon form of A, 1.
pivot column – A column of A that contains a pivot position.
the span of a set of vectors (i.e. Span {v1, v2, v3, …, vn})
The collection of all vectors that can be written in the form c1v1+ c2v2+ ⋯ + cpvp with c1, ⋯, cp scalars.
Glossary Deff. – Span {v1, ⋯ ,vp }: The set of all linear combinations of v1, ⋯,vp. Also, the subspace spanned (or generated) by v1,⋯,vp.
The vector space made up of all linear combinations of the set {v1, v2, v3, … , vn}.
homogeneous system of linear equations
A system of linear equations that can be written in the form Ax = 0, where A is an m x n matrix and 0 is the zero vector in Rm.
trivial solution to a homogeneous system
Ax = 0, has at least one solution, namely, x=0(the zero vector in Rm.
parametric vector form of a solution to Ax = b.
x = su + tv (s, t ∈ R).
s & t are the weights
u & v are the vectors
|x1 | |-1 | + |4|
x = |x2| = |2 | + x3 |0|
|x3| = |2 | + |1|
A solution in the form of x = xp + xh, where xh is a linear combination. h = the homogeneous solution and p = the particular solution.
set of linearly independent vectors
An indexed set of vectors {v1, v2, v3, …, vn}) in Rn with a vector equation: c1v1+ c2v2 + ⋯ + cpvp = 0. Where the weights are all zero.
set of linearly dependent vectors
If there exist weights c1, c2, …, cp, not all zero, such that:
{v1, v2, v3, …, vn}) in with a vector equation: c1v1+ c2v2 + ⋯ + cpvp = 0.
linear transformation (or function or mapping) T from Rn to Rm.
A rule that assigns to each vector x in Rn a vector T(x) in Rm. Must meet the following properties:
- T(x + y) = T(x) + T(y)
- T(cx) = cT(x)
domain, co-domain, and range of a linear transformation of a Rn to Rm transformation
domain - The set Rn.
co-domain - The set Rm.
range - The set of all images T(x).
The image of vector x under the action of a linear transformation T
T(u) = Au = | 3 5 | • | 2 | = | 1 |
| -1 7| | -1 | | -9 |
1 -3| | 5 |
contraction, dilation, shear, projection, rotation
contraction – T: R2 → R2 by T(x) = rx when 0 ≤ r ≤ 1.
dilation – T: R2 → R2 by T(x) = rx when r > 1
shear: a transformation that changes one portion of a point (x, y) such as x, without changing y.
projection:
x1 | 1 0 0 | | x1 | = | x1 |
x2 →| 0 1 0 | | x2 | = | x2 |
x3 | 0 0 0| | x3 | = | 0 |
rotation:
A = | cos σ - sin σ |
| sin σ cos σ |
standard matrix for a linear transformation
The matrix A such that T(x) = Ax.
onto transformation
A mapping T: Rn → R<span>m</span> is said to be onto Rm if each b in Rm is the image of at least one ** x** in Rn.
Many points in the domain can map to the same point in the range.
one-to-one transformation
A mapping T: Rn → Rm is said to be one-to-one if each b in Rm is the image of at most one x in Rn. Each point in the domain maps to a unique point in the range.
transpose of a matrix
Given an m x n matrix, the transpose of A is the n x m matrix, denoted by AT, whose columns are formed from the corresponding rows of A.
C = | 1 1 1 1 |
| -3 5 -2 7 |
CT = | 1 -3 |
| 1 5 |
| 1 -2 |
| 1 7 |
invertible matrix
A matrix with the following attributes:
- An n x n matrix A is said to be invertible if there is an n x n matrix C such that CA = 1 & AC = 1.
- A matrix A where A-1 exists.
- A matrix A that is row equivalent to In.
- A • A-1 = In
singular(non-invertable) and non-singular (invertable)
singular - a matrix in Rn (the opposite of each of the items below applies to non-singular)
- pivots < n (free variables).
- Ax = 0 has more that one solution.
- Not invertable.
- Not row equivalent to I.
- The columns of A do not span Rn.
- The columns of A form a linearly dependent set.
- AT is singular.
- T(x) = Ax is not a one-to-one function.
- T(x) = Ax does not map Rn onto Rn
- There exist some b in Rn, Ax = b is not consistent.
- There is not an n by n matrix C such that CA = 1.
- detA = 0.
elementary row operation
There are three:
- (Replacement) Replace one row by the sum of itself and a multiple of another row.
- Interchange) Interchange two rows.
- (Scaling) Multiply all entries in a row by a nonzero constant.
elementary matrix
A matrix that is obtained by performing a single elementary row operation on an identity matrix.
subspace of Rn
any set H in that has three properties:
- The zero vector is in H.
- For each u and v in H, the sum u + v is in H.
- For each u in H and each scalar c, the vector cu is in H.
column space of A
the set Col A of all linear combinations of the columns of A. The colunm b such that Ax = b has a solution.
null space of A
the set Nul A of all solutions to the homogeneous equation Ax = 0.
basis of a subspace of Rn
A linearly independent set in H that spans H.
RREF the matrix and then put in parametric vector form.
Must:
- Be linearly independent
- Span the subspace
coordinate vector of x relative to a basis B
the coordinates of x ralative to the basis ß are the weights c1, … , cp such that x = c1b1 + … + cpbp, and the vector in Rp
[x]ß = |c1|
|...|
|c<sub>p</sub>|
is called the coordinate vector of x relative to a basis ß
dimension of a subspace
the number of bectors in any basis for H.
rank of a matrix (denoted by rank A)
the dimension of the column space of A. (# of columns)
spanning (or generating) set of a subspace
a sent {v1, … , vp} in H such that H = Span {v1, … , vp}.
kernel of a linear transformation (or null space)
the set if all u in V such that T(u) = 0. (the zero vector in W)
eigenvalue ŷ of a square matrix
A scalar ŷ(should be upside-down y) is called an eigenvalue of A if there is a nontrivial solution x of Ax =ŷx; such an **x **is called an eigenvector corresponding to ŷ.
eigenvector associated with ŷ(should be upside-down y)
a x such that x is a nontrivial solution of Ax =ŷx
A nonzero vector x such that Ax =ŷ**x **for some scalar ŷ.
characteristic equation of a square matrix
The scalar equation det(A - ŷI) = 0
ŷ (should be upside-down y)
The Invertible Matrix Theorem
If A is an invertible matrix.
A-1 exists
The Invertible Matrix Theorem
A is/isn’t row equivalent to the n x n identity matrix.
A is row equivalent to the n x n identity matrix.
The Invertible Matrix Theorem
If A is an m x n matix, A has ___ pivot positions.
A has n pivot positions.
The Invertible Matrix Theorem
The equation Ax=0 has ___________ solution.
The equation Ax=0 has only the trivial solution.
The Invertible Matrix Theorem
The columns of A for a linearly _________ set.
The columns of A for a linearly independent set.
The Invertible Matrix Theorem
The linear transformation x |→ Ax is/isn’t one-to-one.
The linear transformation x |→ Ax is one-to-one.
The Invertible Matrix Theorem
The equation Ax = b has ________ solution(s) for each b in Rn.
The equation Ax = b has at least one solution for each b in Rn.
The Invertible Matrix Theorem
The columns of A ________ Rn.
The columns of A span Rn.
The Invertible Matrix Theorem
The linear transformations x |→ Ax ________ Rn onto Rn.
The linear transformations x |→ Ax maps Rn onto Rn.
The Invertible Matrix Theorem
There is an n x n matrix C such that CA = ____.
There is an n x n matrix** **C such that CA = I.
The Invertible Matrix Theorem
There is an n x n matrix D such that AD=______.
There is an n x n matrix D such that AD=I.
The Invertible Matrix Theorem
AT is/isn’t an invertible matrix.
AT is an invertible matrix.
Determinate of an n x n matrix A
Computed by a cofactor expansion across any row or down any column.
The Invertible Matrix Theorem
The columns of A form a ______ of Rn.
The columns of A form a basis of Rn.
The Invertible Matrix Theorem
Col A = R<span>?</span>.
Col A = Rn.
The Invertible Matrix Theorem
dim Col A = _______
dim Col A = n
The Invertible Matrix Theorem
rank A = _____
rank A = n
The Invertible Matrix Theorem
Nul A = {?}
Nul A = {0}
The Invertible Matrix Theorem
dim Nul A = ______
dim Nul A = 0
The rank of matrix A
Denoted by rank A - Is the dimension of the column space of A.
Rank is equal to the number of pivot columns in Reduced Row Echelon Form.
If A is a 2 x 2 matrix with zero determinant, then one column of A is a ___________ of the other
Multiple
If two rows of a 3 x 3 matrix A are the same, then det A = ______.
0
If A is a 3 x 3 matrix, then det 5A _ 5 det A.
Not equal
If A and B are n x n matrices, with det A = 2 and det B = 3, then det(A + B) _ 5.
Not eqaul.
If A is n x n and det A = 2, then det A3 _ 6.
Not equal.
If B is produced by interchanging two rows of A, then det B _ det A.
Not equal, det B = -det A.
If B is produced by multiplying row 3 of A by 5, then det B _ 5 det A.
Equals.
If B is formed by adding to one row of A a linear combination of the other rows, then det B _ det A.
Equals.
det AT _ -det A
False. det AT = det A.
det(-A) _ -det A.
Not equals.
detATA _ 0.
>=, Greater than or equals to.
Any system of n linear equations in n variables can be solved by Cramer’s rule - T/F
False
If u and v are in R2 and det [u v] = 10, then the area of the triangle in the plane with vertices at 0, u, and v is __.
10.
If A3 = 0, then det A = __.
0, zero.
If A is invertable, then det A-1 _ det A.
Does not equal.
det A-1 = 1/det A
If A is invertable, then (det A)(det A-1) _ 1.
Equals
dimPn=
Where P is a polynomial such x2+1 = P2
dimPn=n+1
T is a Linear Transformation if
- T(u + v) = T(u) + T(v)
- T(cu) = cT(u)
basis of column space
The pivot columns of a matrix A form a **basis **for the column space of A.
basis of nul space
The pivot columns of the homogeneous solution after RREF.
basis of row space
The pivot rows of the modified matrix A after RREF
kernel
Same as nul Space
dimension of the vector space
The # vectors in the basis or the # of pivots.
dimmensions of the nul Space
rankA
= dimA
The Rank Theorem
For m x n matrix:
- rank A + dim Nul A = n
- common dimension = rank A = # of pivots = dim of row space.
Let ß = {b1, … , bn} and C = {c1, … , cn} be bases of a vector space V. Then there is a unique n x n matric cPß such that
[x}c = cPß [x]ß
The unique n x n matric cPß is known as:
The change-of-coordinates matric from ß to C such that:
[x}c = cPß [x]ß
