Vector Spaces and Subspaces
Strip vectors down to the two operations and the axioms they satisfy, so that polynomials, matrices and functions count as vectors too. A subspace is a subset closed under both operations.
What a learner can do afterwards
- Check the closure conditions to decide whether a subset is a subspace
- Give an example of a vector space whose elements are not arrows
- Show that the solutions of a homogeneous system form a subspace
1 · Read
A vector space is a set with an addition and a scaling that obey the standard rules. Arrows qualify, but so do many other kinds of objects. Matrices of one shape are an example: add two of them entry by entry, and scale one by multiplying every entry by the same number. Check closure under both operations first: sums and scalings must stay inside the set. The remaining rules usually ride along from the parent space.
All polynomials of degree at most 3 form a vector space, and none of them is an arrow. Adding or scaling them never reaches degree 4, and the zero polynomial qualifies. Note the bound says at most: polynomials of degree exactly 3 miss zero and fail. Likewise, symmetric matrices qualify: in each one, the entry in row i, column j equals the entry in row j, column i. Invertible matrices do not, since a matrix plus its own negative is the singular zero matrix.
A subspace is a slice that is itself a vector space. Demand zero first, then test the two closures: sums of members stay inside, and scalings stay inside. The x-axis passes everything, while the line y equals x plus 1 misses the origin and fails at once. The first quadrant fails scaling, since minus 1 times (1, 1) escapes it.
A homogeneous system has all its constants zero: A times x equals 0. Its solutions always form a subspace, and linearity is why. If A times x and A times y are both zero, then so are A times their sum and A times any scaling. But solutions of A times x equals b with nonzero b never form a subspace: zero is missing, since A times zero is zero, not b.
A vector space closes under addition and scaling, a subspace must also hold zero, and homogeneous solutions always qualify.
Reading notes describe three open resources listed on OER Commons: Vector Spaces First (University of Ottawa), Linear Vector Space, and Linear Algebra: An Interactive Introduction (Ximera). Practice questions are original.
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14 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.