Questions and Answers
Define Linear Transformation and its properties.
A mapping from Rn to Rm denoted by T is linear if T(u+v) = Tu + Tv, T(αv) = αT(v), T(0) = 0, and T(au + bv) = aT(u) + bT(v) for every u, v ∈ Rn.
What is the standard matrix A?
Te1, Te2 * (x1@x2) = Ax.
What is the set Rn?
The domain of T.
What is the set Rm?
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How is T completely determined?
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Define a projection matrix.
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Define the image of x.
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What is a shear transformation?
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Linear systems are expressed in terms of what?
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Vector equations are viewed as a?
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Describe the mapping view.
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Explain the view of the matrix equation A = (1&-1&2@2&1&1), x = (1@1@1), b = (0@0).
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How does Ax=b lead to a transformation?
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What is the precise solution set of Ax=b?
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In particular, N(A) is...
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Describe a matrix-vector multiplication.
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What is the pre-image of a matrix A of size m by n?
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What is the image of x?
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Describe the range of A and how we get it.
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What is the image of b = (1@2) for A = (1&-1&2@2&1&1), x = (1@0@0)?
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How do you find a pre-image?
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Can you determine a particular solution of this matrix equation by inspection: A = (1&-1&2@2&1&1), x = (x@y@z), b = (2@4)?
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What is the pre-image of (2@4) for A = (1&-1&2@2&1&1), x = (x@y@z)?
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Express N(A) as a span when A = (1&-1&2@2&1&1).
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What is the range of A for A = (1&-1&2@2&1&1)?
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Describe the first geometric example in linear transformations.
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Describe the second geometric example in linear transformation.
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Study Notes
Linear Transformations Overview
- A linear transformation maps vectors from Rn to Rm and is denoted as T.
- Properties of linear transformations include:
- T(u + v) = Tu + Tv (additivity)
- T(αv) = αT(v) (homogeneity)
- T(0) = 0 (maps the zero vector to zero)
- T(au + bv) = aT(u) + bT(v) for scalars a, b.
Standard Matrix Representation
- The standard matrix A is defined by the action of T on the standard basis vectors e1 and e2, represented as: Te1, Te2 * (x1@x2) = Ax.
Vector Spaces
- Rn represents the domain of the linear transformation T.
- Rm is the co-domain of T.
Determining T
- T is completely determined by its effects on the columns of the identity matrix In.
Projection Matrix
- A projection matrix B is symmetric and satisfies B^2 = B, exemplified by B = v*v^T, where v is a unit vector.
Image and Shear Transformations
- The image of a vector x is the resulting vector b in Rm after applying T.
- A shear transformation involves fixing one coordinate while shifting others: T(x) = Ax.
Linear Systems and Vector Equations
- Linear systems can be expressed through vector equations (Ax = b).
- Vector equations can be understood in a "mapping" context, where matrix A transforms vector x into vector b.
Mapping View
- Viewing Ax = b as a mapping illustrates how matrix A acts on vector x to output vector b. This dynamic perspective emphasizes relationships between input and output vectors.
Matrix Equations and Transformations
- A matrix equation can be interpreted as a transformation from Rn to Rm, mapping pre-images (x) in the domain to images (b) in the co-domain.
- The null space, N(A), is the set of all vectors in Rn that are sent to the zero vector.
Range of A
- The range of matrix A is the collection of all images Ax produced for vectors from Rn, representing the output space of the transformation.
Finding Pre-images and Solutions
- To determine a pre-image, solve the linear system, find a particular solution, and add elements from the null space of A.
Geometric Interpretation
- The first geometric example involves mapping from R^3 to R^3, where A = (1&0&0@0&1&0@0&0&0) projects onto the xy-plane, transforming coordinates (x,y,z) to (x,y,0).
Further Exploration
- The second geometric example remains to be defined, inviting further inquiry into geometric implications of linear transformations.
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Explore essential definitions and properties of linear transformations through this interactive flashcard quiz. Each card addresses key concepts, including the standard matrix representation and the linearity criteria. Perfect for anyone studying linear algebra.