Questions and Answers
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.
True
If T: R2 to R2 rotates vectors about the origin through an angle theta, then T is a linear transformation.
True
When two linear transformations are performed one after another, the combined effect may not always be a linear transformation.
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A mapping T: Rn to Rm is onto Rm if every vector x in Rn maps onto some vector in Rm.
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If A is a 3 x 2 matrix, then the transformation X to Ax cannot be one-to-one.
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Not every linear transformation from Rn to Rm is a matrix transformation.
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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.
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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.
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A mapping T: Rn to Rm is one-to-one if each vector in Rn maps onto a unique vector in Rm.
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If A is a 3 x 2 matrix, then the transformation X to Ax cannot map R2 onto R3.
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What is the condition for a mapping T: Rn to Rm to be said to be onto Rm?
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What is the condition for a mapping T: Rn to Rm to be said to be one-to-one?
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When is a linear transformation T: Rn to Rm said to be one-to-one?
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Under what condition does T: Rn to Rm map onto Rm?
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If a linear transformation T: Rn to Rm maps Rn onto Rm, must m be greater than or equal to n?
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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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Test your knowledge with these flashcards covering key concepts from Linear Algebra Chapter 1.9. Each card presents important definitions and theorems regarding linear transformations and matrices. Perfect for students looking to reinforce their understanding of this critical mathematical topic.
