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Mathematics Department Stanford UniversityMath 51H Mid-Term 1

October 14, 2009

Unless otherwise indicated, you can use resultscovered in lecture and homework, provided they are clearly stated.

If necessary, continue solutions on backs of pagesNote: work sheets are provided for your convenience, but will not be graded

75 minutes

Q.1Q.2Q.3Q.4

T/20

Name (Print Clearly):

I understand and accept the provisions of the honor code (Signed)

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Name: Page 1/4

1 (a) (3 points): Suppose

A D

0BB@

1 0 2 1 12 1 1 3 10 0 0 1 12 0 4 0 4

1CCA

Find (i) a basis for the null spaceN.A/

ofA

(show all row operations!), and (ii) a basis for the columnspace C.A/.

Make sure you justify your results by referring to the appropriate results from lecture.

1 (b) (2 points): Give the definition of lim an D `, where {an}nD1;2;::: is a given sequence in R and` 2 R, and use your definition to prove ` 0, assuming that the limit ` exists and that an 08n.

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2 (a) (3 points): Suppose v1; : : : ; vk are non-zero vectors in Rn and let m. k/ be the maximum

value of the set of integers ` 2 {1; : : : ; k} such that it is possible to select ` l.i. vectors from thegiven vectors v1; : : : ; vk. Prove that any collection ofm l.i. vectors selected from v1; : : : ; vk mustautomatically provide a basis for span{v1; : : : ; vk}.

(b) (2 points): Suppose that S is a non-empty bounded subset ofR. Prove that there is a sequence{an}nD1;2;::: with an 2 S for each n D 1;2; : : : and liman D supS.

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3 (a) (3 points): Let V be a subspace ofRn. Define the orthogonal projection PV W Rn ! Rn ofRn

onto V and give the proof that x 2 Rn ) kx PV.x/k kx yk8y 2 V.

(b) (2 points): Prove that 2jx yj kxk2 C kyk2 for all vectors x; y 2 Rn.

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4 (a) (3 points): Suppose A is an mn matrix and b 2 Rm. (i) Give the proof that Ax D b has at leastone solution x 2 Rn b 2 C.A/, and (ii) In case m D n and N.A/ D {0}, prove that Ax D b hasa solution for each b 2 Rn.

4 (b) (2 points): Suppose A D .aij/ is an m n matrix and x 2 Rn. Give the proof of the inequality

kAxk kAkkxk, where kAk DqPm

iD1

PnjD1 a

2ij.

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work-sheet 1/2

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work-sheet 2/2

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