PracticeTest

# PracticeTest - ∈ R n × n is the equation A B 2 = A 2 2...

This preview shows pages 1–4. Sign up to view the full content.

Linear Algebra Midterm Summer 2011 1. Consider the system of equations x 1 + 2 x 2 + 3 x 3 + 2 x 4 = 5 4 x 1 + 8 x 2 + x 3 + x 4 + 6 x 5 = 10 3 x 1 + 6 x 2 + x 3 + 2 x 4 + 5 x 5 = 15 2 x 1 + 4 x 2 + x 3 + 9 x 4 + 10 x 5 = 30 This can be succintly written as A x = b and represented in matrix form as [ A | b ] R 4 × 6 . Given that rref([ A | b ]) = 1 2 0 0 0 27 0 0 1 0 0 - 18 0 0 0 1 0 16 0 0 0 0 1 - 16 (a) Find ker A as a span of vectors. (You’ll need to consider rref([ A | 0 ]) here.) (b) Find im A as a span of vectors. (c) Find a particular solution s of the system A x = b . 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
(d) Use parts (a) and (c) to ﬁnd the set of all possible solutions K = s + ker A , and write this in the form K = { s + t 1 v 1 + ··· + t k v k | t 1 ,...,t k R } where the vectors v i are the basis vectors of ker A . (e) Use part (d) with t 1 = ··· = t k = 3 to ﬁnd a second solution s 2 of A x = b and verify that this is indeed a solution. 2. Let A R n × n satisfy A 2 = A . Show that if all entries of A are nonzero, then A is not invertible. [Hint: It’s easier to prove the contrapositive of this statement.] 2
3. If a and b are real numbers, we know that ( a + b ) 2 = a 2 + 2 ab + b 2 . If A,B

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ∈ R n × n , is the equation ( A + B ) 2 = A 2 +2 AB + B 2 still true? If so, prove it, if not, ﬁnd examples of matrices A and B for which this fails. 4. Let A = 2 4 8 4 5 1 7 9 3 . (a) Find ker A and im A as spans of vectors. 3 (b) Find all solutions of the system A x = b , where b = 6 9 16 . 5. Consider the basis β = ±² 1 1 ³ , ² 1 2 ³´ for R 2 . Find the representation [ x ] β of the vector x = ²-2 5 ³ in this basis. 6. Let ρ = { e 1 , e 2 } be the standard basis for R 2 and let β = ±² 1 1 ³ , ² 1 ³´ be another basis for R 2 . If T A ∈ L ( R 2 , R 2 ) is the counterclockwise rotation through π/ 4, with associated matrix A = " 1 √ 2-1 √ 2 1 √ 2 1 √ 2 # ﬁnd the matrix representation [ T A ] ρ,β of T A in these bases. 4...
View Full Document

{[ snackBarMessage ]}

### Page1 / 4

PracticeTest - ∈ R n × n is the equation A B 2 = A 2 2...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online