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Math 205 Exam 1 - February 3, 2006, Exams of Linear Algebra

Bidhan Chandra Krishi ViswavidyalayaLinear Algebra

The solutions manual for exam 1 of math 205, a college-level linear algebra course. It includes problems on finding parametric vector solutions of a system of linear equations, determining the image and injectivity of a linear transformation, finding the standard matrix of a linear transformation, solving linear systems, calculating matrix inverses, and proving properties of vectors and matrices. The document also includes problems on the dot product and linear independence.

Typology: Exams

2012/2013

Uploaded on 02/27/2013

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NAME:
Math 205 - Exam 1 - February 3, 2006
1. (8 pts.) Describe all solutions of Ax=0in parametric vector form, where the matrix Ais
1−4−2 0
2−8 1 3
0 1 −1 3
.
How can these solutions be described geometrically?
2. (8 pts. each) Consider the transformation T:R4→R3, given by T(x) = Ax, where Ais row equivalent
to
1−4 8 1
0 2 −1 3
0 0 0 5
.
(a) Does Tmap R4onto R3? Explain.
(b) Is Tone-to-one? Explain.
pf3
pf4
pf5

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NAME:

Math 205 - Exam 1 - February 3, 2006

  1. (8 pts.) Describe all solutions of Ax = 0 in parametric vector form, where the matrix A is  

How can these solutions be described geometrically?

  1. (8 pts. each) Consider the transformation T : R^4 → R^3 , given by T (x) = Ax, where A is row equivalent to  

(a) Does T map R^4 onto R^3? Explain. (b) Is T one-to-one? Explain.

  1. (8 pts.) Let T : R^2 → R^2 be a linear transformation that first reflect points through the vertical x 2 -axis and then reflects points through the line x 2 = x 1. Find the standard matrix of T.
  2. (4 pts. each) Consider the linear system

x 1 + hx 2 = 2 3 x 1 + 6 x 2 = k

For what value(s) of h and k (if any) does this system have

(a) no solutions? (b) a unique solution? (c) infinitely many solutions?

  1. (8 pts.) The dot product of two vectors u and v in Rn, denoted by u · v, is defined by u · v = uT^ v. For

v =

[

]

, find v · v. How does this compare to the distance from the point (0, 0) to the point (3, 4)? (Hint: Draw a right triangle and use the Pythagorean theorem..)

  1. (8 pts.) Suppose A is a 3x4 matrix and y is a vector in R^3 such that the equation Ax = y does not have a solution. Does there exist a vector z in R^3 such that the equation Ax = z has a unique solution? Explain your answer.
  1. (8 pts.) Let T : Rn^ → Rm^ be a linear transformation, and let {v 1 , v 2 , v 3 } be a linearly dependent set of vectors in Rn. Explain why the set {T (v 1 ), T (v 2 ), T (v 3 )} is linearly dependent.
  2. (8 pts.) Suppose A is an invertible n × n matrix. Explain, without using the invertible matrix theorem, why the columns of A are linearly independent.

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