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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) Matrixvector 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 onetoone?  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 :
Norow 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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