Connect abstract maps with matrices, kernels and images.
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Try
A linear map can preserve some information and erase other information. T(x,y,z)=(x+z,2y) maps three input coordinates to two outputs. Inputs (1,0,0) and (0,0,1) coincide after mapping, revealing a lost direction.
Predict
Predict whether T(x)=Ax+b is always linear. It is linear only when the fixed offset b is zero. Linearity means T(au+bv)=aT(u)+bT(v) for all vectors and scalars; in particular T(0)=0. The kernel contains inputs mapped to zero; the image contains outputs actually reached.
Experiment
For the displayed T, compute T(e₁), T(e₂), T(e₃) and use them as columns to construct [[1,0,1],[0,2,0]]. Find a nonzero input sent to zero. Then construct an input for an arbitrary target (r,s), for example (r,s/2,0).
Schematic domain-to-codomain diagram for T(x,y,z)=(x+z,2y). Distinct inputs (1,0,0) and (0,0,1) have the same output (1,0); the diagram illustrates lost information, not geometric distance.
Understand
Worked argument: T(x,y,z)=0 requires x+z=0 and 2y=0, so ker T is the line spanned by (−1,0,1). Every target (r,s) has the input just constructed, so the image is all R². Kernel and image belong to different spaces.
More generally a linear map is determined by its values on a basis. If x=Σxⱼeⱼ, then T(x)=ΣxⱼT(eⱼ), explaining the standard matrix columns. The same argument works with polynomial bases: differentiating a+bt+ct² gives b+2ct, a linear map from P₂ to P₁.
Invalid test: showing T(0)=0 proves linearity. Squaring satisfies that condition but fails additivity, since (1+1)²≠1²+1².
Apply
Verify additivity and scaling together through the linear-combination identity. To describe an image, give all reachable outputs and demonstrate that each proposed output is reached. To describe a kernel, solve the zero-output equation completely.
Continue
Depth: injectivity is equivalent to a zero kernel: equal outputs imply their difference lies in the kernel. Surjectivity means the image equals the declared codomain. Enlarging the codomain can change surjectivity without changing the rule itself.
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Practice and reasoning
Drafts save locally as you type. Reveal worked solutions when useful; the site remembers that exposure. Automatic matches check the requested answer, while written arguments require review against the rubric.
For T(x,y,z)=(x+z,2y), enter the first coordinate of T(3,−1,4).
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
For T:R³→R², T(x,y,z)=(x+z,2y), find dim ker T.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Which map R→R is linear?
The jth column of the standard matrix of T:Rⁿ→Rᵐ is:
Prove ker T is a subspace of the domain for a linear map T.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Let D:P₂→P₁ send a polynomial to its derivative. Show D is linear and determine a basis for its kernel and image.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
