Linear Algebra Answers

1. Let S be the subspace of R⁵ defined by S = { (x₁, x₂, x₃, x₄, x₅) E R⁵ : x₁ = x_2, x₃ = 2x₄ + x₅ }. Then the dimension of S is
2. Let T: R³ -> R³ be defined as T (x,y,z) = (x+y, x-y, x+2z). Then the basis of range T is...
3. Which of the following transformations are linear:

(i) T₁ : R³ -> R² by T₁ (u, v, w) = ( u - v + 2w, 5v - w).

(ii) T₂ : P (R) -> R by T₂ (P) = (integral sign from b to a) 2p(x)dx for a,b E R with a<= b

(iii) T₃ : P(R) -> P(R) by T₃ (P(U) = UP (U) + U

4.. Suppose T : R² -> M₂₂ is a linear defined by T (u,v) = [u v]

[u 2u]

then Ker (T) is...

5.. Suppose T: R⁶ -> R⁴ is a linear map such that null T = U where U is a 2-dimentional subspace of R⁶ . Then dim range T is...

6.. For a given 2x2 matrix A = [ 5 -3 ]

[ -6 2]

the matrix P that is diagonalizes A can be written as P = ...

1. What is the basis for the null space of the set "\\begin{vmatrix}\n 1 &2& 1 \\\\\n 1 &1 & 0\\\\\n-1 & 1 & 0\\\\\n1 & 4 & 1\n\\end{vmatrix}"
2. For a given matrix A = "\\begin{bmatrix}\n 5 & -3 \\\\\n -6 & 2\n\\end{bmatrix}" , the matrix P that is diagonalizes A is

(i) P = "\\begin{bmatrix}\n 1 & 1 \\\\\n 2 & -1\n\\end{bmatrix}"

(ii) P = "\\begin{bmatrix}\n 1 & -1 \\\\\n -2 & 1\n\\end{bmatrix}"

(iii) P = "\\begin{bmatrix}\n 1 & 0 \\\\\n 0 & 1\n\\end{bmatrix}"

3.. Suppose T:R² -> M₂₂ is a linear defined by T(u,v) = "\\begin{bmatrix}\n u & v \\\\\n u & 2u\n\\end{bmatrix}" then Ker (T) is ...

1. Suppose T: R³ -> R³ is linear and has an upper-triangular matrix with respect to the basis (1, 0, 0), (1,2,1), (1,2,2). Then the orthonormal basis of R³ with respect to which T has an upper-triangular matrix is...
2. For u = (1,2,2) and v= (1, -2, -1), the value of ||u-v|| is...
3. Suppose that u,v E V, where V is a real vector space such that ||u|| = 4 and ||v|| = 3 Then < u + v, u - v > is...
4. In R³, let U = Span ((1, 0, 0), (0, 1/sqrt2 , 1/sqrt2 )) The u E U such that || u - ( 2,4,6)|| is as small as possible is...
5. Let T: R³ -> R³ defined as T(x,y,z) = (2x, x+y, x-z). Then adjoint operator T* (u,v,w) is...

(i) (2u+v+w, u, -w)

(ii) (2u,v+w, u-w)

(iii) (u,v, -w)

1. Let S be the subspace of R⁵ defined by S = { (x₁, x₂, x₃, x₄, x₅) E R⁵ : x₁ = x_2, x₃ = 2x₄ + x₅ }. Then the dimension of S is
2. Let T: R³ -> R³ be defined as T (x,y,z) = (x+y, x-y, x+2z). Then the basis of range T is...
3. Which of the following transformations are linear:

(i) T₁ : R³ -> R² by T₁ (u, v, w) = ( u - v + 2w, 5v - w).

(ii) T₂ : P (R) -> R by T₂ (P) = (integral sign from b to a) 2p(x)dx for a,b E R with a<= b

(iii) T₃ : P(R) -> P(R) by T₃ (P(U) = UP (U) + U

4.. Suppose T : R² -> M₂₂ is a linear defined by T (u,v) = [u v]

[u 2u]

then Ker (T) is...

5.. Suppose T: R⁶ -> R⁴ is a linear map such that null T = U where U is a 2-dimentional subspace of R⁶ . Then dim range T is...

6.. For a given 2x2 matrix A = [ 5 -3 ]

[ -6 2]

the matrix P that is diagonalizes A can be written as P = ...

1. Let S be the subspace of R⁵ defined by S = { (x₁, x₂, x₃, x₄, x₅) E R⁵ : x₁ = x_2, x₃ = 2x₄ + x₅ }. Then the dimension of S is
2. Let T: R³ -> R³ be defined as T (x,y,z) = (x+y, x-y, x+2z). Then the basis of range T is...
3. Which of the following transformations are linear:

(i) T₁ : R³ -> R² by T₁ (u, v, w) = ( u - v + 2w, 5v - w).

(ii) T₂ : P (R) -> R by T₂ (P) = (integral sign from b to a) 2p(x)dx for a,b E R with a<= b

(iii) T₃ : P(R) -> P(R) by T₃ (P(U) = UP (U) + U

4.. Suppose T : R² -> M₂₂ is a linear defined by T (u,v) = [u v]

[u 2u]

then Ker (T) is...

5.. Suppose T: R⁶ -> R⁴ is a linear map such that null T = U where U is a 2-dimentional subspace of R⁶ . Then dim range T is...

6.. For a given 2x2 matrix A = [ 5 -3 ]

[ -6 2]

the matrix P that is diagonalizes A can be written as P = ...

During an experiment in the lab, Jack, a biologist has a 40% mixture and a 10%mixture of the nutrient that he got from the pawpaw plant. The lecturer posed the question to Jack. How much of each mixture should Jack mix to obtain 25 cm³ of a 28% mixture? [Verify your answer by MATHEMATICA and attach the printout of the commands and output]

The subraction of a matrix B may be considered as the addition of the matrix(-1)B.Does the commutative law of addition permit us to state that A-B=B-A?If not,how would you correct the statement?

Suppose A is a matrix with characteristic polynomial p("\\lambda" ) ="\\lambda"^3 - "\\lambda"

a) What is the order of the matrix A?

b) Is A invertible?

c) Is A diagonalisable?

d) Find the eigenvalues of A²