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Unit Vector - Linear Algebra - Quiz, Exercises of Linear Algebra

Kumaun UniversityLinear Algebra

This is the Quiz of Linear Algebra which includes Zero Vector, Linearly Dependent, Statement, Vector, Linear Combination, Expressed, Trivial Solution, Inspection, Dependent, Theorem etc. Key important points are: Unit Vector, Direction, Orthogonal Basis, Linear Algebra, Techniques, Formulas Developed, Orthogonal Bases, Vectors Horizontally, Familar Ground, Vectors Vertically

Typology: Exercises

2012/2013

Uploaded on 02/27/2013

senapathy_101
senapathy_101 🇮🇳

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Math 205B&C 03/27/09 Quiz 07 page 1 Name 8am 1:10 pm
1. Suppose that v1=
3
−4
5
,v2=
3
1
−1
and v3=
1
x
y
; let s=
−11
18
51
.
1a. Explain why v1⊥v2.
1b. Find a unit vector in the direction of v2.
1c. Find xand ywhich make B={v1,v2,v3}an orthogonal basis of R3. (Use good linear algebra
techniques; your answer will involve a RREF).
1d. Use the formulas developed in class for orthogonal bases to find α2for which s=α1v1+α2v2+α3v3.
(You do not have to find α1and α3.)
2. If A=
233 2
121 3
215−6
211 2
then RREF(A)isR=
100 1
010 2
001−2
000 0
Find a basis for each of the
following. Write vectors horizontally where appropriate.
2a. Col(A)2b. Col(R)
2c. Row(A)2d. Row(R)
2e. Express r3(ie, row 3) of Aas a linear combination r3=xr1+yr2+zr4. of the other three rows of
A. (Hint: you will be on familar ground if you write the vectors vertically to solve the problem; find xy
and z. Or explain why there are no such scalars. Use good linear algebra methods)

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Math 205B&C 03/27/09 Quiz 07 page 1 Name 8am 1:10 pm

  1. Suppose that v 1 =

, v 2 =

 (^) and v 3 =

x y

; let s =

1a. Explain why v 1 ⊥ v 2.

1b. Find a unit vector in the direction of v 2.

1c. Find x and y which make B = {v 1 , v 2 , v 3 } an orthogonal basis of R^3. (Use good linear algebra techniques; your answer will involve a RREF).

1d. Use the formulas developed in class for orthogonal bases to find α 2 for which s = α 1 v 1 +α 2 v 2 +α 3 v 3. (You do not have to find α 1 and α 3 .)

  1. If A =

 then RREF(A) is^ R^ =

 Find a basis for each of the

following. Write vectors horizontally where appropriate.

2a. Col(A) 2b. Col(R)

2c. Row(A) 2d. Row(R)

2e. Express r 3 (ie, row 3) of A as a linear combination r 3 = xr 1 + yr 2 + zr 4. of the other three rows of A. (Hint: you will be on familar ground if you write the vectors vertically to solve the problem; find x y and z. Or explain why there are no such scalars. Use good linear algebra methods)