Linear Algebra Answers

Find the orthogonal projection of u on a
u = (6, 6, -5) and a = (1, -2, 0)

Give the exact answer, using fractions if necessary.
projau = (?, ?, ?)

Thank you for your help!

which method of solving systems of equations is best for you to understand and explain why (substitution or addition). Add an example to your post so other students could understand what you mean.

ex:
2x-3y=6
3x+6y=16

x/2 + y/3 = 2/3
x/2 + 2/y = -1
What is the solution of the system?

my school work is online. I can not copy and paste. How would you all be able to help me? There is a possibility of log back in & it being new question...

Suppose A is an m x n matrix and B is an n x l matrix. Further, suppose that A has a row of zeros. Does AB have a row of zeros? Why or why not? Does this also hold true if B has a row of zeros? Why or why not?


Give an example of two matrices A and B for which AB=0 with A not equal to 0 and B not equal to 0.

If T is a transpose map, such that T(A) = transpose of A for
any 2 by 2 matrix A with real entries, find the eigenvalues of T and a basis of M_2(R) (where M_2(R) denote all the 2 by 2 matrices with real entries) with respect to which T is diagonal

6x^2 + 3x - 18 = 0

Suppose M= [1,0;1,1]
Explain why the function T(x)= Mx maps the x-axis onto the line y=x and why it maps the line y=2 onto the line y=x+2 by calculating M[t,0] and where the image points lie and similarily M[t,2]

Using Cramer's rule, solve the following system of linear equations:

x + 4y - 2z = 3
x + 3y = 1 - 7z
2x = 8 - 9y - z

Can you explain how to do it? Help please!

Let β={u1, u2, ... , un} be a subset of Fn containing n distinct vectors and let B be an nxn matrix in F having uj as column j.

Prove that β is a basis for Fn if and only if det(B)≠0.

For one direction of the proof this is something that I found :

Since β consists of n vectors, β is a basis if and only if these vectors are linearly independent, which is equivalent to the map LB being one-to-one. Since the matrix B is square, this is in turn equivalent to B being invertible, hence having a nonzero determinant.

However I do not understand the transition from the vectors being linearly independent to being one to one. Why is this true? Is there a better way to prove this? Also, how do I prove the reverse direction?

How do I solve this equation using the subtitution method:


60+0.20(x-550)=120+0.15(x-800)