Matrix Practice Test 1
Practice Test 1
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,
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
1. Is the given transformation linear ? Why or why not?
(a) Is T invertible? Why or why not? --- Yes, it's 2 × 2 and
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 v3 = 5v1 × 3v3
(a) Is A invertible? --- No - it has only two pivots (see
(b) What is rank(A)? --- 2
(c) What is nullity(A)? --- 1
6. Let in the span of
Yes, via row row reduction of the corresponding augmented matrix.
(a) What is the span of S? --- The row reduction above gives
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
Give a matrix representing T.
(a) Are the columns of T linearly independent ? --- Yes, the
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
9. Show that the inverse of
Use the row reduction method