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What is the condition for a unique matrix A in a linear transformation T: Rn to Rm?
What is the condition for a unique matrix A in a linear transformation T: Rn to Rm?
There exists a unique matrix A such that T(x) = Ax for all x in Rn.
A linear transformation T: Rn maps onto Rm is completely determined by its effects on the columns of the n x n identity matrix.
A linear transformation T: Rn maps onto Rm is completely determined by its effects on the columns of the n x n identity matrix.
True (A)
If T: R2 to R2 rotates vectors about the origin through an angle theta, then T is a linear transformation.
If T: R2 to R2 rotates vectors about the origin through an angle theta, then T is a linear transformation.
True (A)
When two linear transformations are performed one after another, the combined effect may not always be a linear transformation.
When two linear transformations are performed one after another, the combined effect may not always be a linear transformation.
A mapping T: Rn to Rm is onto Rm if every vector x in Rn maps onto some vector in Rm.
A mapping T: Rn to Rm is onto Rm if every vector x in Rn maps onto some vector in Rm.
If A is a 3 x 2 matrix, then the transformation X to Ax cannot be one-to-one.
If A is a 3 x 2 matrix, then the transformation X to Ax cannot be one-to-one.
Not every linear transformation from Rn to Rm is a matrix transformation.
Not every linear transformation from Rn to Rm is a matrix transformation.
The columns of the standard matrix for a linear transformation from Rn to Rm are the images of the columns of the n x n identity matrix.
The columns of the standard matrix for a linear transformation from Rn to Rm are the images of the columns of the n x n identity matrix.
The standard matrix of a linear transformation from R2 to R2 that reflects points through the horizontal axis, the vertical axis, or the origin takes the form [a 0; 0 d] where a and d are +/- 1.
The standard matrix of a linear transformation from R2 to R2 that reflects points through the horizontal axis, the vertical axis, or the origin takes the form [a 0; 0 d] where a and d are +/- 1.
A mapping T: Rn to Rm is one-to-one if each vector in Rn maps onto a unique vector in Rm.
A mapping T: Rn to Rm is one-to-one if each vector in Rn maps onto a unique vector in Rm.
If A is a 3 x 2 matrix, then the transformation X to Ax cannot map R2 onto R3.
If A is a 3 x 2 matrix, then the transformation X to Ax cannot map R2 onto R3.
What is the condition for a mapping T: Rn to Rm to be said to be onto Rm?
What is the condition for a mapping T: Rn to Rm to be said to be onto Rm?
What is the condition for a mapping T: Rn to Rm to be said to be one-to-one?
What is the condition for a mapping T: Rn to Rm to be said to be one-to-one?
When is a linear transformation T: Rn to Rm said to be one-to-one?
When is a linear transformation T: Rn to Rm said to be one-to-one?
Under what condition does T: Rn to Rm map onto Rm?
Under what condition does T: Rn to Rm map onto Rm?
If a linear transformation T: Rn to Rm maps Rn onto Rm, must m be greater than or equal to n?
If a linear transformation T: Rn to Rm maps Rn onto Rm, must m be greater than or equal to n?
If T is one-to-one, must m be greater than or equal to n?
If T is one-to-one, must m be greater than or equal to n?
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Study Notes
Linear Transformations and Matrices
- A linear transformation T: Rn → Rm corresponds to a unique matrix A such that T(x) = Ax for any x in Rn.
- The effect of linear transformation is determined by how it acts on the columns of the identity matrix I.
Characteristics of Linear Transformations
- Rotating vectors about the origin by an angle θ in R2 is an example of a linear transformation.
- The combination of two linear transformations is always a linear transformation, preserving vector addition and scalar multiplication.
Mapping Conditions
- A mapping T: Rn → Rm is defined to be onto Rm if for each vector in Rm there exists at least one vector in Rn that maps to it.
- A transformation is one-to-one if every vector in Rn maps to a unique vector in Rm.
Matrix Dimensions and Transformations
- A 3 x 2 matrix cannot create a one-to-one transformation due to insufficient dimensions.
- If A is a 3 x 2 matrix, it cannot map R2 onto R3 because the output dimension must match or exceed the input dimension.
Properties of Standard Matrices
- The columns of the standard matrix for a linear transformation from Rn to Rm reflect the images of the identity matrix columns.
- Standard matrices for reflections (horizontal, vertical, origin) are structured in a specific format, where entries can only be ±1.
Linear Independence and Spanning
- For a linear transformation to be onto Rm, the columns of the corresponding matrix A must span Rm.
- A transformation is one-to-one if the columns of A are linearly independent, ensuring that each input maps to a unique output.
Dimension Relations
- To have a transformation T: Rn → Rm that is onto, the matrix A must have as many columns as rows (m ≥ n).
- If T is one-to-one, the matrix A must contain a pivot in every column, leading to the same dimensional requirement (m ≥ n).
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