Midterm 2 Review

Question 1

How to check if a subset is a subspace of R3

Answer 1

1. Does it have elements (Is the 0 vector possible?)
2. If vector u and vector v are in s then so its u +v
3. If u vector is in s then Ku is in S

Question 2

Basis for Col(A)

Answer 2

all columns in RR matrix that have a pivot and then write the corresponding in OG matrix

Question 3

Steps for finding Null(A)

Answer 3

1. Write out the system of equation = 0
2. Solve in terms of free variables
3. In vector parametric form, write the coeffs of those FV
4. The vectors are the basis (REMEMBER TO PUT IN SET NOTATION)

Question 4

Is this a subspace of R3 and its a span of vectors

Answer 4

Yes since every span is a subspace

Question 5

One-to-one

Answer 5

Does the homogenous equation have FV in RR matrix?

Question 6

Onto

Answer 6

Does every row have a pivot?

Question 7

How to find the determinant of a 2x2 matrix

Answer 7

Cross-multiply the diagonals :
ad - bc

Question 8

Operations in RR and How They Affect Determinants

Answer 8

The only thing that affects the determinant is if you scale it by a constant.

Question 9

How to find a domain?

Answer 9

What space does the vector lie in? AKA how many entries in the vector

Question 10

How to find the codomain?

Answer 10

What space do the outputs lie in?

Question 11

How to check is linear

Answer 11

Can you add u + v and get the same result

Question 12

Standard Matrix and Linear Transformations: Expansion/Contradiction

Answer 12

Changes the area

Question 13

Standard Matrix and Linear Transformations: Reflection (y-axis)

Answer 13

Folded over y-axis

Question 14

Standard Matrix and Linear Transformations: Reflection (x-axis)

Answer 14

Folded over x-axis

Question 15

Standard Matrix and Linear Transformations: Rotations

Answer 15

Changes the angle around the origin like a clock

Question 16

Standard Matrix and Linear Transformations: Horizontal Shearing

Answer 16

Push out horizontally but height stays the same

Question 17

Standard Matrix and Linear Transformations: Vertical Shearing

Answer 17

Changes the angle but not the area

Question 18

Standard Matrix and Linear Transformations: projection/shearing x-axis

Answer 18

Everything is now on the x-axis line

Question 19

Standard Matrix and Linear Transformations: projection/shearing y-axis

Answer 19

Everything is now on the y-axis line

Question 20

The first three Invertible Matrix Theorem

Answer 20

1. There is a matrix where XA=I
2. There is a matrix X where AX = I
3. A transposed is invertible

Question 21

Middle Two Invertible Matrix Theorem

Answer 21

1. A has exactly n pivots
2. A row reduces to an identity matrix

Question 22

Middle three Invertible Matrix Theorem

Answer 22

1. T (linear transformation) is one-to-one
2. Ax=0 has only a trivial solution
3. If the column of A are linearly independent

Question 23

Last Three Invertible Matrix Theorem

Answer 23

1. T is onto
2. Ax=b is always consistent
3. Range of T = co-domain