Linear Algebra Answers

find the eigenvalue and eigenvector for matrix


0 0 5
S= -2 1 1
3 0 2

Suppose that {v1.....Vn} is a basis for the vector space V. Given any vector v element of V, we can express v as a linear combination v = X1V1+.....+XnVn. The uniqueness of this expression means that mapping v to the n-tuple of coefficients (X1...Xn) defines a function O: V - R^n. Prove that this function is a bijection.

look at matrix

1 -1
Z = 2 4


write matrix Z in its Jordan form ?

look at the matrix
A =
1 2 -2 -1
0 1 2 -3
1 4 2 -7
1 3 0 -4

what is the dimension of the row space of the matrix A ?
what is the dimension of the column space of the matrix A ?
what is the dimension of the null space of the matrix A ?
Give the null space for matrix A ?

look at vectors
1 1 0
u= 0 v= 1 w = 1
1 0 -1

what is the dimension of the space spanned by these three vectors ?

1
is the vector -1 in tyhe span of vectors u , v , and w ?
4

If A is an 7 x 7 matrix with all 49 entries being odd numbers. Show that det (A) is a multiple of 64. You may use the fact (without proving) that an n x n matrix with integer entries has an integer determinant.

If X and Y are n*n matrices, Show that (I-XY) is invertible if and only if (I-YX) is
invertible.
Hint: Use X(I-YX)=(I-XY)X

Assess what the results of the regression equation tells managers and how it is likely to impact decisions made related to maximizing profitability.

matrix[2x_5y]+matrix[x-y_-6x]=matrix[6_2]

solve the following equation for x and y.

Let K be a division ring with center k. Show that any ideal I ⊆ R has the form R • h where h ∈ k[x].