Vector and Matrix Operations : Row Vectors, Column Vectors, Dot Product and Echelon Form
1. Find the dot product for the following pairs of vectors:
a. Row vector = (2 0) Column vector is below
5
18
b. Row vector = (3 9 -4) Column vector is below
3
0
2
c. Row vector = (5 6 7 8)
1
1
1
1
The following matrices will be used in problems 2-3 below:
0 -2 5
A= 3 -4 17
1 2 3
9 7 2
-3 3 0
B= -4 -5 9
-2 0 6
-8 6 4
3 3 0 -7
X= 7 5 1 1
4 3 3 -1
4 1 0 5
Y= 3 -2 7 1
-5 8 -6 2
2. Answer True or False using the matrices above
a. 2A + 2B = 2(A + B)
b. AT + BT = (A + B)T
c. Let C = A * X, then C will be a 3 by 3 matrix
d. Let D = Y * B, then D will be a 3 by 3 matrix
e. (A * X)T = AT * BT
3. Do the specified matrix multiplications. Show your work.
a. A * X
b. Y * B
c. A * Y
d. B * X
4. Find the inverses of the following matrices
a.
8 6
5 4
b.
3 2
7 4
c.
8 5
-7 -5
d.
3 -4
7 -8
5. Use elementary row operations to reduce the following matrices to echelon or reduced echelon form
a.
0 -2 -1
3 0 0
-1 1 1
b.
3 5 4
1 0 1
2 1 1
c.
3 6 7
0 2 1
2 3 4
Please see the attached file for the fully formatted problems.
Vector and Matrix Operations, Row Vectors, Column Vectors, Dot Product and Echelon Form are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.
Question (1)
Find a basis and dimension of the subspace W of R4 generated by the vectors
( 1 , - 4 , - 2 , 1 ) , ( 1 , - 3 , - 1 , 2 ) , ( 3 , - 8 , - 2 , 7 ) .
Extend it to find the basis of R4 .
Question (2)
Determine a basis and the dimensions of the Subspace of M2(R) generated by the
2 by 2 Matrices [ 2 -10 ] , [
Please find the least squares solution of linear system Ax=b, and find the orthogonal projection of b onto the column space of A.
I) Matrix 3*2 A=[1 1;-1 1;-1 2] , b=[7;0;7]
II) A=[2 0 -1;1 -2 2;2 -1 0;0 1 -1] , b=[0;6;0;6]
Answers: I) x1=5, x2=1/2, [11/2;-9/2;-4] II) x1=14, x2=30, x3=26; [2;6;-2;4]
Find the ortho
Question (1):
Let T: R^3 into R^3 be a linear transformation defined by
T(x, y, z) = (x + 2y - z, y + z, x + y - 2z )
Find a basis and the dimension of ( i ) Range of T ( ii) the Kernel of T
Question (2) :
If T:R^4 into R^3 is a linear transformation defined by
T( a, b, c, d) = ( a - b + c + d, a + 2c - d,
See the attached file.
The objective is to write MATLAB codes that calculate scalar, vectorand triple (scalar) products of vectors.
Example: the scalar product function is R^3
Copy and paste the following programme into the MATLAB editor and save the program as scalar_prod.m
function z = scalar_prod(x,y)
Please see the attached file for the fully formatted problems.
Compute the products using the rowvector rule for computing Ax. If a product is undefined, explain why.
1) 2) 3)
4) let A = and b = . Show that the equation Ax=b does not have a solution for all possible b, and describe the set of all b for which Ax=b does
(1) a. Find the (vector) equation of the plane passing through the points (1,2,-2),
(-1,1,-9), (2,-2,-12).
b. Find the (vector) equation of the plane containing (1,2,-1) and perpendicular to
(3,-1,2).
(2) Suppose a, b, c are non zero vectors.
a. Explain why (a x b) x (a x c)
