Question

#2. For the matrix A = 1 2 1 2 3 7 4 7 9 find the following. (a) The null space N (A) and a basis for N (A). (b) The range space R(AT ) and a basis for R(AT )

. #3. Consider the vectors −→x = k − 6 2k 1 and −→y = 2k 3 4 . Find the number k such that the vectors −→x and −→y are orthogonal (perpendicular.)

#4. Find the number k such that the vectors −→a = −8 2k 2 and −→b = 2k −8 −2 are co-linear.

Answer #1

LINEAR ALGEBRA
For the matrix B=
1 -4 7 -5
0 1 -4 3
2 -6 6 -4
Find all x in R^4 that are mapped into the zero vector by the
transformation Bx.
Does the vector:
1
0
2
belong to the range of T? If it does, what is the pre-image of
this vector?

(a) Do the vectors v1 = 1 2 3 , v2 = √ 3 √ 3 √ 3 , v3=√ 3 √ 5 √
7, v4 = 1 0 0 form a basis for R 3 ? Why or why not?
(b) Let V ⊂ R 4 be the subspace spanned by the vectors a1 and
a2, where a1 = (1 0 −1 0) , a2 = 0 1 0 −1. Find a basis for the
orthogonal complement V ⊥...

Do the vectors v1 = 1 2 3 ,
v2 = √ 3 √ 3 √ 3 ,
v3 √ 3 √ 5 √ 7 ,
v4 = 1 0 0 form a basis for R 3 ? Why or why not?
(b) Let V ⊂ R 4 be the subspace spanned by the vectors a1 and
a2, where a1 = ...

. Given the matrix A =
1 1 3 -2
2 5 4 3
−1 2 1 3
(a) Find a basis for the row space of A
(b) Find a basis for the column space of A
(c) Find the nullity of A

A (–4, –1, 2), B (3, –2, –1) and C (–1, 3, –4),
AB= 7? − ? − 3?
CB = 4? − 5? + 3?
AC = 3? + 5? - 2?
Question 7: Express the vector AC as the sum of two vectors: AC
= ? + ?, where ? is parallel to the vector CB and ? is
perpendicular to CB. Given that AC ∙ CB = −26 and that CB = √50,
determine the y-component of...

2. Find a basis for R 4 that contains the vectors X = (1, 0, 0,
1) and Y = (1, 1, 0, 1).
Note: I'm not looking for the orthogonal basis, im looking for a
basis that contains those 2 vectors.

True or False
If A is the matrix of a projection onto a line L in R 2 and the
vector x in R 2 is not the zero vector, then the vector x − Ax is
perpendicular to the vector x.
If vectors u, v, x and y are vectors in R 7 such that u = 2v +
0x − 3y, then a basis for span(u, v, x, y) is {u, v, y}.

8. Consider a 4 × 2 matrix A and a 2 × 5 matrix B .
(a) What are the possible dimensions of the null space of AB?
Justify your answer.
(b) What are the possible dimensions of the range of AB Justify
your answer.
(c) Can the linear transformation define by A be one to one?
Justify your answer.
(d) Can the linear transformation define by B be onto? Justify
your answer.

Let A be a 2x2 matrix
6 -3
-4 2
first, find all vectors V so the distance between AV and the
unit basis vector e_1 is minimized, call this set of all vectors
L.
Second, find the unique vector V0 in L such that V0 is
orthogonal to the kernel of A.
Question: What is the x-coordinate of the vector V0 equal to.
?/?
(the answer is a fraction which the sum of numerator and
denominator is 71)

(a) Use the Gram-Schmidt process on the basis {(1, 2, 2),(1, 2,
3),(4, 3, 2)} of R ^3 find an orthonormal basis.
(b) Write the vector v = (2, 1, −5) as a linear combination of
the orthonormal basis vectors found in part (a).

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