Linear Transformation Given by a Matrix In Exercises 23-28, define the linear transformation
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- Linear Transformation Given by a Matrix In Exercises 23-28, define the linear transformation T:RnRm by T(v)=Av. Use the matrix A to a determine the dimensions of Rn and Rm, b find the image of v, and c find the preimage of w. A=[121101], v=(5,2,2), w=(4,2)arrow_forwardThe Standard Matrix for a Linear TransformationIn Exercises 1-6, find the standard matrix for the linear transformation T. T(x,y)=(5x+y,0,4x5y)arrow_forwardThe Standard Matrix for a Linear TransformationIn Exercises 1-6, find the standard matrix for the linear transformation T. T(x,y,z)=(x+y,xy,zx)arrow_forward
- Finding the Standard Matrix and the Image In Exercises 11-22, a find the standard matrix A for the linear transformation T, b use A to find the image of the vector v, and c sketch the graph of v and its image. T is the reflection in the y-axis in R2: T(x,y)=(x,y), v=(2,3).arrow_forwardFinding the Standard Matrix and the Image In Exercises 11-22, a find the standard matrix A for the linear transformation T, b use A to find the image of the vector v, and c sketch the graph of v and its image. T is the reflection in the line y=x in R2: T(x,y)=(y,x), v=(3,4).arrow_forwardFinding the Standard Matrix and the Image In Exercise 11-22, a find the standard matrix A for the linear transformations T, b use A to find the image of the vector v, and c sketch the graph of v and its image. T is the projection onto the vector w=(3,1) in R2:T(v)=2projwv, v=(1,4).arrow_forward
- Finding the Kernel of a Linear Transformation In Exercise 1-10, find the kernel of the linear transformation. T:P3P2T(a0+a1x+a2x2+a3x3)=a1x+2a2x2+3a3x3arrow_forwardLinear Transformation Given by a Matrix In Exercises 33-38, define the linear transformations T:RnRmby T(v)=Av. Find the dimensions of Rnand Rm. A=[1213400210]arrow_forwardFinding an Image and a PreimageIn Exercises 1-6, find a the image of v and b the preimage of w for the linear transformation. T:R2R2, T(v1,v2)=(v1+v2,2v2), v=(4,1), w=(8,4).arrow_forward
- Finding the Standard Matrix and the Image In Exercises 11-22, a find the standard matrix A for the linear transformation T, b use A to find the image of the vector v, and c sketch the graph of v and its image. T is the counterclockwise rotation of 45 in R2, v=(2,2).arrow_forwardLinear Transformation Given by a Matrix In Exercises 33-38, define the linear transformations T:RnRmby T(v)=Av. Find the dimensions of Rnand Rm. A=[012114500131]arrow_forwardWriting For the linear transformation from Exercise 34, find a T(2,4), b the preimage of (1,2,2) c Then explain why the vector (1,1,1) has no preimage under this transformation. Linear Transformation Given by a Matrix In Exercises 33-38, define the linear transformations T:RnRm by T(v)=Av. Find the dimensions of Rn and Rm. A=[122422]arrow_forward
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