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Spring 2011 - Math 461

Midterm #1 - Practice Exam

1. [10pts] In each of the following, justify your answer carefully. Unjustified answers will

receive no credit.

(a) Let A be a n× n matrix such that the equation Ax =  0 only has the trivial solution.Is A invertible?

(b) Let A be a 3 × 4 matrix and  b be a vector in R3 that is not in the set spanned by thecolumns of A. Can A have a pivot in every row?

(c) Let T  : R2−→ R

3 be a linear transformation. Can T  be onto? Can T  be one-to-one?

2. [20pts] Find all the solutions of the following linear system of equations (write your answer

in parametric vector form):

x1 − 2x4 = −32x2 + 2x3 = 0

x3 + 3x4 = 1−2x1 + 3x2 + 2x3 + x4 = 5

3. [20pts] Consider the following matrix:

B=

1 3 −1 −2−

1−

5 5 80 1 −2 h ,

where h is a real parameter.

(a) For what value(s) of h do the columns of B span R3? Justify your answer.

(b) For what value(s) of h does the equation Bx =  0 have a non-trivial solution? Justify

your answer.

4. [15pts] Let A =

 a1  a2  a3  a4

be a 3× 4 matrix, which is row equivalent to the matrix

1 4 −1 5

0 2 4 −30 0 0 5

(a) Find the reduced echelon matrix which is row equivalent to A.

(b) Are the columns of  A linearly dependent? If so, find a linear dependence relationamong the columns of A.

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5. Let T  : R2→ R

2 be a linear transformation that transforms the vector e1 into

1−3

and

the vector e2 into

2−4

.

(a) [5pts] Find the standard matrix of T .

(b) [5pts] Find the image of the vector c =

1−2

under T .

(c) [5pts] Find a linear transformation S  : R2→ R

2 such that S (T (x)) = x for all x (S  iscalled the inverse of T , denoted T −1).

6. [20pts] Is the following matrix invertible? If so, find its inverse:

1 −2 1

3 −3 00 2 −1