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Linear Algebra
6x^2 + 3x - 18 = 0
Linear Algebra
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]
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]
Linear Algebra
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!
x + 4y - 2z = 3
x + 3y = 1 - 7z
2x = 8 - 9y - z
Can you explain how to do it? Help please!
Linear Algebra
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?
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?
Linear Algebra
How do I solve this equation using the subtitution method:
60+0.20(x-550)=120+0.15(x-800)
60+0.20(x-550)=120+0.15(x-800)
Linear Algebra
Suppose x, y are elements of R^n and are nonparallel vectors.
a) Prove that if sx + ty = 0, then s = t = 0.
b) Prove that if ax + by = cx + dy, then a = c and b = d.
a) Prove that if sx + ty = 0, then s = t = 0.
b) Prove that if ax + by = cx + dy, then a = c and b = d.
Linear Algebra
Let a,b and c be three vectors such that a + b + c = 0. Given that a x b = b x c = c x a. Explain what this means geometrically. (Your answer should refer to a triangle)
Linear Algebra
Let A=1/3 (-2 -1 2
2 -2 1
1 2 2)
Prove that A is the product of a rotation and a reflection. Prove that A is an orthogonal matrix.
2 -2 1
1 2 2)
Prove that A is the product of a rotation and a reflection. Prove that A is an orthogonal matrix.
Linear Algebra
B=(x[sup]2[/sup]+x, x[sup]2[/sup]-2, x[sup]2[/sup]+2x-1) is a subset of the vector space P2 of polynomials of degree no larger than two,T(x[sup]2[/sup]+x) =(1,-2) T(x[sup]2[/sup]-2) = (4,1)& & T(x[sup]2[/sup]+2x-1) = (2,-1)what is matrix representation for T with respect to the bases B for P2 and S=(1,0)(0,1) for R[sup]2[/sup]?
With T given as in the above question, calculate T(7x[sup]2[/sup]+3x-2).
With T given as in the above question, calculate T(7x[sup]2[/sup]+3x-2).
Linear Algebra
Consider the set of all functions [0,1] which are discontinuous. check if this set is a vector space over R with respect to pointwise addition and scalar multiplication
