# Matrix Practice Test 1

# Practice Test 1

## Exam Topics

1. De nitions:

(a) Vector Space

(b) Linear Transformation

(c) Linear Independence /Dependence

(d) Linear Combination

(e) Matrix-vector and Matrix- matrix multiplication

(f) Domain, codomain, null space, range, column space, nullity, rank,

span

2. (2 × 2) Rotation Matrices

3. Elementary Row Operations /Elementary Matrices

4. Systems of Linear Equations/Solution by Row Reduction (including

existence and uniqueness of solutions )

5. REF vs. RREF

6. Matrix Inversion, Theorem 2.6

## Practice Test

1. Is the given transformation linear ? Why or why not?

Yes

2. Let

(a) Is T invertible? Why or why not? --- Yes, it's 2 × 2 and
one

column is not a multiple of the other .

(b) Are the columns of T linearly independent ? Why or why not? ---

Yes, because it's invertible

(c) What is the rank of T? --- 2

(d) What is the nullity of T? --- 0

3. Let Write b as a
linear combi -

nation of the vectors in S.

4. Give the general solution in vector form to the equation Ax = b, where

(a) Is A one-to-one? --- no

(b) Is A onto? --- no

(c) What is rank(A)? --- 2

(d) What is nullity(A)? --- 1

(e) Are the columns of A linearly independent ? --- no

5. Determine if the following vectors are linearly independent :

No|row reduction would give two pivots, or v_{3} = 5v_{1} × 3v_{3}

(a) Is A invertible? --- No - it has only two pivots (see
above)

(b) What is rank(A)? --- 2

(c) What is nullity(A)? --- 1

6. Let in the span of
S?

Yes, via row row reduction of the corresponding augmented matrix.

(a) What is the span of S? --- The row reduction above gives
three

pivots, thus the span is R^3

(b) If is A invertible? --- Yes, every column
is

a pivot column.

(c) What are rank(A) and nullity(A)? --- rank(A) = 3, nullity(A) = 0

7. Find the inverse of

Use A^{-1} to solve the system of equations

x_{1} + 2x_{2} + 3x_{3} = 1

2x_{1} + 3x_{2} + 4x_{3} = 2

3x_{1} + 4x_{2} + 6x_{3} = 3

If Ax = b, then x = A^{-1}b, so

8. Suppose

Give a matrix representing T.

(a) Are the columns of T linearly independent ? --- Yes, the
matrix is

2 × 2 and one column is not a multiple of the other.

(b) What are the rank and nullity of T? --- 2,0

(c) Is T invertible? Why or why not? --- Yes, the columns are linearly

independent.

9. Show that the inverse of

Use the row reduction method

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