MAST10007 Chap.8 Linear Transformations and Standard Matrices
Linear Transformations and Standard Matrices
Chapter 8 develops linear transformations and standard matrices for University of Melbourne MAST10007. A transformation is linear when it preserves vector addition and scalar multiplication. In finite-dimensional standard coordinates, every linear map is multiplication by a matrix, whose columns are the images of the standard basis vectors. Composition reads right to left: if S acts first and T second, the matrix is [T][S].
This order is the same reason matrix multiplication is not generally commutative. It pairs exact definitions with a new completed example, two concept diagrams, diagnostic contrasts and answered retrieval prompts.
A standard matrix is built from what the transformation does to basis directions, so each column has a meaning before any general vector is multiplied.
The linearity conditions let those basis images determine every other output. Composition must follow the action order, and a proposed inverse must undo the transformation on the relevant space. Geometry can often be read from columns, but the claim should be checked algebraically through images, products, or preserved quantities.
This keeps rotations, reflections, projections, and shears connected to the same matrix framework.
What this chapter covers
- 01
Transformation language
- 02
The two linearity conditions
- 03
Consequences of linearity
- 04
Standard matrices
- 05
Images of basis vectors
- 06
Rotations and reflections
- 07
Projections and shears
- 08
Composition order
- 09
Inverses of transformations
- 10
Reading geometry from columns
- 11
Building columns from standard-basis images
- 12
Tracking action order through composition
- 13
Checking inverse transformations by composition
Linear Transformations and Standard Matrices worked example
- stepApply T to e1=(1,0) to get (1,3), which becomes the first column.
- stepApply T to e2=(0,1) to get (2,-1), which becomes the second column.
- stepThus A=[[1,2],[3,-1]]. Multiply A(2,-1) to obtain (0,7).
- stepDirect substitution into the rule gives (2-2,6+1)=(0,7), confirming the matrix and column order.
Key terms
- Transformation language
- A transformation assigns an output to each input. Its domain and codomain are part of its definition. The range may be smaller than the codomain, so those words must not be used interchangeably.
- The two linearity conditions
- Linearity requires T(u+v)=T(u)+T(v) and T(cu)=cT(u) for every permitted vector and scalar. Both conditions matter. Combining them gives T(au+bv)=aT(u)+bT(v), the engine behind basis representations.
- Consequences of linearity
- Every linear map sends zero to zero and preserves all finite linear combinations. These are consequences, not substitutes for a full linearity proof unless a counterexample is being used to show a map is not linear.
- Standard matrices
- The standard matrix of T:R^n→R^m has T(e1),…,T(en) as its columns. Its size is m×n. Multiplying by a coordinate vector automatically forms the correct linear combination of those image columns.
- Images of basis vectors
- Images of basis vectors determine a linear map completely. For a nonstandard basis, first express the input in that basis or use the representation matrix with clearly labelled input and output coordinate systems.
- Rotations and reflections
- Rotations and reflections are recognised by what they do to basis directions and by length preservation. A rotation matrix has orthonormal columns and determinant 1; a planar reflection has determinant −1.
- Composition order
- If one transformation acts first and another acts second, the corresponding matrix product places the second transformation's matrix on the left. Reading a composition from right to left preserves the actual order in which vectors are transformed.
- Inverse transformation
- An inverse transformation reverses the original map on both its domain and codomain. In matrix form, composing in either order produces the appropriate identity. Existence requires the map to be one-to-one and onto between the relevant spaces.
Linear Transformations and Standard Matrices FAQ
How do I check transformation language?
A transformation assigns an output to each input. Its domain and codomain are part of its definition. The range may be smaller than the codomain, so those words must not be used interchangeably. No, because T(0,0)=(1,0), not zero.
What is the main trap in linear transformations and standard matrices?
A map with a nonzero constant translation is not linear because T(0)≠0. Columns are images of basis vectors, not rows. For composition, the transformation performed first appears on the right.
Can AI help with this MAST10007 topic?
Yes. Ask Sia to explain one step, create a fresh practice problem, or check your reasoning. Use it to learn rather than complete graded assessment, and follow University of Melbourne academic-integrity rules.
Why are standard-matrix columns the images of basis vectors?
Every vector is a linear combination of the standard basis vectors. Linearity carries that combination to the same combination of their images, and matrix multiplication uses input coordinates to form precisely that column combination.
How can I detect a reversed composition?
Write the action sequence on a generic input before multiplying matrices. The rightmost matrix must act first. You can also test a basis vector through the transformations one at a time and compare the result with the proposed product's corresponding column.
Exam move
Use the varied 9-page chapter as a dependency map: define each object, reproduce the fresh worked example, explain both diagrams, answer the two retrieval prompts, and perform an independent check. For linear transformations and standard matrices, revisit the first line where your representation, dimensions, or theorem conditions diverge.
Build each standard matrix from labelled basis images before applying it to general inputs. For composition, draw arrows showing which map acts first and let that sequence determine product order. Verify a result twice: once by direct use of the transformation rule and once by matrix multiplication. For geometric maps, explain what the columns do to the coordinate directions before naming the geometry.
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