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We are given that
T(v) = Av
where A =
We need to define T(v)
Let v = , then T(v) = Av =
Which reflects v about the x_{2}axis.
If the linear transformation T(v) = Av rotates the vectors (1, 0) and (0,1), x radians clockwise, then:
We are given that the linear transformation T(v) = Av rotates the vectors (1,0) and (0,1), π radians clockwise.
We need to find the matrix A.
We know that if a vector (a, b) is rotated through an angle a under T. Then
Therefore, the matrix of T is
A =
If the linear transformation T(v) = Av rotates the vectors v_{1 }= (1, 0) and v_{2} = (0, 1) clockwise π radians, the resulting vectors are:
We are give that the linear transformation T(v) = Av rotates the vectors v_{1} = (1,0) and v_{2} = (0,1) clockwise π radians. We need to find the image of v_{1 }= (1, 0) and v_{2} = (0,1) under T
We know that if a vector (a, b) is rotated through an angle α under linear transformation T, then
Here v_{1} = (1, 0) and a = π
Therefore,v_{2} = (0,1)
= (1,0)
and T(v_{2}) = T(0, 1)
= (0,1)
If the linear transformation T(v) = Av rotates the vectors v_{1} = (1, 0) and v_{2} = (0, 1) clockwise π/2 radians, the resulting vectors are:
We are given that the linear transformation T(v) = Av rotates the vector v_{1} = (1, 0) and v_{2} = (0,1) clockwise π/2radians.
We know that if a vector (a, b) is rotated through an angle a clockwise direction under linear transformation T.Then
Here, v_{1 }= (1 , 0) and a = π/2 radians
Therefore, T(v_{1}) = T(1, 0)
= = (0,1)
Therefore, T(1, 0) = (0,1)
Similarly, for v_{2} = (0, 1) and α = π/2,
and T(v_{2}) = T(0 ,1)
= (1,0)
Therefore, T(0,1) = (1 ,0 ).
If the linear transformation T(v) = Av rotates the vectors (1, 0) and (0, 1) clockwise π/2 radians then:
We are given that the linear transformation T(v) = Av rotates the vectors (1, 0) and (0, 1) clockwise π/2 radians. We need to find the matrix A.
We know that if a vector (a, b) is rotated through an angle a clockwise under T. Then
Here, (a, b) = (1, 0) and a = π/2
Therefore, T (1, 0)
= (0, 1) = 0(1, 0) + 1(0, 1)
and (a, b) = (0,1), α = π/2
= (1, 0) =  1(1, 0) + 0(0, 1)
Therefore, the matrix of T is A =
If A = satisfies the matrix equation A^{2}  kA + 2I = 0, then what is the value of k?
We need to find the value o f k where the matrix
A = satisfies the equations A^{2}  kA + 2I = 0.
But A =
So,
Putting these values in the equation
A^{2}  kA + 2I= 0, we get
or
or 1  3k + 2 = 0 => k = 1
Under which one of the following condition does the system of equations have a unique solution?
We are given that the system of equations
Reduce this system of equations to echelon form, using the operations "R_{2} > + R_{2}  2R_{1}" and R_{3} > R_{3 } R_{1}.
These operations yields:
Also we have, this system has a unique solution so a  8 ≠ 0
or a ≠ 8
We need to find the determinant of the given matrix,
determinant of A =
or det (A) =
det (A) = 1/n!
If the system of equations
x  2y  3z = 1, (p + 2)z = 3, (2p + 1)y + z = 2 is inconsistent, then what is the value of P?
We are given that the system of equations,
x  2y  3z = l, (p + 2)z =3, (2p + l) y + z = 2 is inconsistent. Then we need to find the value of p.
The augmented matrix of this system of equations is,
The solution of this matrix is inconsistent if
p + 2 = 0
or p = 2
If x, y, z are in AP with common difference d and the rank of the matrix is 2, then the value of d and k are
We are given that x, y, z are in A.P. with common difference d i. e.,
or
and the rank of is z.
We need to find the values of d and k. The given matrix can be written as,
Applying R_{2} > R_{2}, R_{1} and R_{3 }> R_{3 }R_{2
}
Applying, R_{3} > R_{3}  R_{2},
We have
or equivalently
T(x,y) = (x,y)
The linear transformation T(x, y) = (x + 2y, x  2y), can be written as a matrix transformation T(x, y) where:
We are given that a linear transformation defined by
T(x, y) = (x + 2y, x  2y)
Let B ={(1, 0), (0, 1)} be a standard basis then
T(1,0) = (1,1) = 1 (1 ,0)+ 1(0,1)
T(0, 1) = (2 ,2) = 2(1,0)  2(0 ,1)
Therefore, the matrix of T with respect to the basis B is
We are given that a transformation defined by
T(x,y,z) = (x,y, 0)
Let α and β be scalars and (x_{1}, y_{1}, z_{1}) and (x_{2}, y_{2}, z_{2}) be two vectors. Then
T[α(x_{1}, y_{1}, z_{1}) + β(x_{1}, y_{2}, z_{2})]
or equivalently
Hence, T is linear.
We are given that a linear transformation T defined by
T(v) = Av
where
Let Then
Let v_{1}, v_{2}, v_{3} ∈ ker T. Then
Using the definition of T, we get
or equivalently
or equivalently
v_{1} = v_{2 }= v_{3} =  v_{1}
Therefore,ker T= {(v_{1}, v_{1},  v_{1}): v ∈ R}.
Therefore, dim ker T= 1
Using RankNullity theorem
Rank T= 3  dim ker T
= 3  1 = 2
Therefore, the range of T is a plane in R^{3}
Suppose T_{1}: V > U and T_{2 }: U —> W be two linear transformation, Then:
We need to find the dim Horn (R^{3}, R^{4}).
Since dim R^{3} = 3 and dim R^{4} = 4
Therefore,dim Horn (R^{3}, R^{4})
= 3 x 4 = 12
Let U and V be n and m dimensional vector space.
Then dim Hom (U, V) = nm
We are given that C is a nonsingular matrix and
B = or B is similar to The eigen values of are all zeroes.
Hence, B^{k} = 0
For k = 3
B^{3} = 0
If X = , the rank of X^{T}X, where X^{T} denotes the transpose of X, is
We are given that the matrix
X=
We need to find the rank of X^{T}X. The transpose of X,
X^{T}=
Now,
Rank (X^{T}X) = 3
The system of equation kx +y + z = 1,x + ky + z = k and x + y + kz = k^{3} does not have a solution, if k is equal to
We are given that the system of equation
does not have a solution. We need to find the value of k. The given linear system of equation can be written as,
This system of equation has no. solution if rank (A) # rank (aug A) the augmented matrix is,
This augmented matrix has no. solutions if k = 2.
We are given that the system of equation
It may be written as
Here, the rank of the given coefficient matrix ≤ 3 and number of unknowns are four.
Hence, rank < no. of unknowns this system of equations has infinitely many solutions.
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