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Quiz 03 for Math 205B: Linear Algebra Problems, Exercises of Linear Algebra

Kumaun UniversityLinear Algebra

A math quiz for math 205b, focusing on linear algebra. It includes four problems with sub-parts, such as finding linear combinations, solutions of equations, and parametric vector forms. Students are expected to use vectors in standard position and matrices to solve the problems.

Typology: Exercises

2012/2013

Uploaded on 02/27/2013

senapathy_101
senapathy_101 🇮🇳

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Math 205B Quiz 03 page 1 01/29/2010 Name
1. Let u,v,z,band kbe as in the following figure. All these vectors belong to R2and all are drawn in “standard
position”.
1A. Find all ways to express bas a linear combination of uand v.
1B. Find all solutions of x1u+x2v=k.
1C. Find one way to write zas a LC of uand v.
1D. Use your answer to 1C to find all solutions of x1u+x2v+x3z=bin parametric vector form x=p+vhwhere pis a
particular solution of x1u+x2v+x3z=band vhrepresents all solutions of the homogeneous equation x1u+x2v+x3z=0.
(Take x3as free). (Hint: From 1A you know a particular solution αu+βv=b. You can add x3(z−z) to the left side of
this because it would be “adding nothing”. Now rewrite the second zas in 1C and simplify).
NOTE: This quiz continues on the other side.
pf2

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Download Quiz 03 for Math 205B: Linear Algebra Problems and more Exercises Linear Algebra in PDF only on Docsity!

Math 205B Quiz 03 page 1 01/29/2010 Name

  1. Let u, v, z, b and k be as in the following figure. All these vectors belong to R^2 and all are drawn in “standard position”.

1A. Find all ways to express b as a linear combination of u and v.

1B. Find all solutions of x 1 u + x 2 v = k.

1C. Find one way to write z as a LC of u and v.

1D. Use your answer to 1C to find all solutions of x 1 u + x 2 v + x 3 z = b in parametric vector form x = p + vh where p is a particular solution of x 1 u + x 2 v + x 3 z = b and vh represents all solutions of the homogeneous equation x 1 u + x 2 v + x 3 z = 0. (Take x 3 as free). (Hint: From 1A you know a particular solution αu + βv = b. You can add x 3 (z − z) to the left side of this because it would be “adding nothing”. Now rewrite the second z as in 1C and simplify).

NOTE: This quiz continues on the other side.

Math 205B Quiz 03 page 2 01/29/2010 Name

  1. Let A =

 and let^ b^ =

2A. Find all solutions of Ax = b and express x in parametric vector form x = p + vh where p is a particular solution of Ax = b and vh represents all solutions of the corresponding homogeneous equation.

2B. It’s a fact that

 is also a solution of^ Ax^ =^ b. What values of the free variables in your answer in (2A) give this

solution?