Linear Algebra Review

  

Review.docx
review.docx

Unformatted Attachment Preview

Don't use plagiarized sources. Get Your Custom Essay on
Linear Algebra Review
Just from $13/Page
Order Essay

Review
?1 1
?1 – 1
1. For the matrix ?
?1 1
?
?3 2
2?
1 6?
?
1 2?
?
5 10?
1
a) Find a basis for the null space.
b) Find a basis for the column space.
2. If
P2
is the vector space of polynomials in t with degree 2 or less and
T : P2 ? R
2
? p(0)?
?
? p(1)?
is defined by T ( p ) = ?
then
a) Prove that T is a linear transformation.
b) Find a basis of kernel T
c) Describe the range of T ?
3. a) Show that
? ?1 ? ?1? ?
B = ? ? ?, ? ? ?
? ? 2? ?1? ?
is basis of
R2
b)
? ? 3? ? 2? ?
2
Given another basis C = ? ? ? , ? ? ? of R , write the change of
? ? 2? ? 2? ?
coordinates matrix
i)
ii)
from
from
B
C
to C
to B
1?
?1 1
?
?
4. Let A= 1 – 1 1
?
?
??1 1 – 1??
i)
ii)
Find the eigenvalues of A.
Diagonalize A, if possible.
Show all your calculations. Do not just write a calculator/computer output.
5. Find an orthonormal basis for the subspace of R 5 spanned by
?? 5 ? ? 3 ? ? 1 ? ?
?? ? ? ? ? ? ?
???- 3? ?- 1? ? 1 ? ??
?? 0 ?, ? 2 ?, ?- 1? ?
?? 0 ? ? 1 ? ? 3 ? ?
?? ? ? ? ? ? ?
???? 0 ?? ?? 1 ?? ?? 2 ?? ??
Show all your calculations. Do not just write a calculator/computer output.
6. Use the Grahm Schmidt process to find an orthogonal basis for the
subspace of C[-1,1] spanned by {1,1+x, 1 + x – 2x 2 }. Use
1
? fgdx
as the
-1
inner product of f and g in C[-1,1].
7. Let A be a square nxn matrix and ? an eigen value of A. Let W=
v ? R n :Av = ?v . Prove that W is a subspace of R n .
?
?
8. Let W be a subspace of a vector space V with a specified inner product.
If W ? contains the vectors of V that are orthogonal to W, prove that W ?
is a subspace of V.
9. Let T : V
spaces
V
V
and
?W
W
Prove that
and
be a one-one linear transformation between the vector
S = ?v1, v2 ,…., vn ? be a linearly independent subset of
?T (v1 ), T (v2 ),…., T (vn )? is a linearly independent subset of
W
i) Write the quadratic form 5 x12 – 4 x1x2 + 5 x2 2 in terms of matrices.
ii)Find a coordinate transformation so that equation 5 x12 – 4 x1x2 + 5 x2 2 -21=0
transforms into an equation without the product term.
10.

Purchase answer to see full
attachment

  
Basic features
  • Free title page and bibliography
  • Unlimited revisions
  • Plagiarism-free guarantee
  • Money-back guarantee
  • 24/7 support
On-demand options
  • Writer’s samples
  • Part-by-part delivery
  • Overnight delivery
  • Copies of used sources
  • Expert Proofreading
Paper format
  • 275 words per page
  • 12 pt Arial/Times New Roman
  • Double line spacing
  • Any citation style (APA, MLA, Chicago/Turabian, Harvard)

Our guarantees

Delivering a high-quality product at a reasonable price is not enough anymore.
That’s why we have developed 5 beneficial guarantees that will make your experience with our service enjoyable, easy, and safe.

Money-back guarantee

You have to be 100% sure of the quality of your product to give a money-back guarantee. This describes us perfectly. Make sure that this guarantee is totally transparent.

Read more

Zero-plagiarism guarantee

Each paper is composed from scratch, according to your instructions. It is then checked by our plagiarism-detection software. There is no gap where plagiarism could squeeze in.

Read more

Free-revision policy

Thanks to our free revisions, there is no way for you to be unsatisfied. We will work on your paper until you are completely happy with the result.

Read more

Privacy policy

Your email is safe, as we store it according to international data protection rules. Your bank details are secure, as we use only reliable payment systems.

Read more

Fair-cooperation guarantee

By sending us your money, you buy the service we provide. Check out our terms and conditions if you prefer business talks to be laid out in official language.

Read more