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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 Ope rations /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. De termine if the following vectors are linearly independent:

No|row reduction would give two pivots , or v3 = 5v1 × 3v3

(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

x1 + 2x2 + 3x3 = 1
2x1 + 3x2 + 4x3 = 2
3x1 + 4x2 + 6x3 = 3

If Ax = b, then x = A-1b, so

8. Suppose

Give a matrix re presenting 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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