MAT 242 Chap.6 Subspaces, Spanning Sets and Linear Independence
Subspaces, Spanning Sets and Linear Independence
From answers to structure
The course notes describe this part of the textbook as its most abstract: the focus moves from computing a single answer to describing a whole set. Rⁿ, the set of column vectors with n entries, is the model.
It is closed under addition and scalar multiplication and satisfies a list of algebraic rules; any set with the same rules is a vector space, and a subset that is a vector space in its own right is a subspace.
The Subspace Test and span
To check that a nonempty subset is a subspace, confirm that sums and scalar multiples of its vectors stay inside it.
Lines and planes through the origin pass; a line that misses the origin fails. Any collection of vectors generates a subspace, its span, which contains every weighted sum of those vectors.
Deciding whether a vector lies in a span means solving a linear system with the spanning vectors as columns and the target after the bar; at least one solution means yes.
Redundancy: dependence and independence
A spanning set may contain vectors that add nothing. The set is linearly dependent if some nontrivial combination equals the zero vector, and independent if only the all-zero weights do.
Row reducing with zeros after the bar settles the question: a column without a pivot means dependence, and the pivot columns identify which original vectors to keep.
The paid chapter answers a membership question, describes an entire span by one equation, and trims a dependent set while keeping the same span.
The calculator cannot reduce a matrix with more rows than columns on some TI models, so the notes suggest padding with zero columns first.
What Test 2 asks
The Test 2 skills list begins with two questions from this chapter: whether a vector lies in the span of given vectors, and whether a set is linearly independent, with a nontrivial combination equal to zero or a reduced spanning set when it is not.
Both are answered by one augmented matrix and its RREF, which the calculator may compute, but the setup must be written out.
Sample solutions penalise heavily the swap of the last column: a target vector belongs after the bar for span, a zero column for independence.
Describing a span by equations
A span can be described in two ways: as all combinations of its spanning vectors, or by the linear equations its members satisfy.
Replacing the target vector by a general vector with letter entries and reducing to echelon form produces those equations in the bottom rows. In R³, the span of two independent vectors is a plane through the origin, described by a single equation in the entries.
What this chapter covers
- 01
Vector spaces and subspaces
- 02
The Subspace Test
- 03
Linear combinations and span
- 04
Deciding membership in a span
- 05
Linear dependence and independence
Worked example · free
Is a vector in the span?
- 2Set up r₁(1, 1, 2) + r₂(0, 1, 1) = (2, 3, 5); the first entry gives r₁ = 2.
- 1The second entry gives 2 + r₂ = 3, so r₂ = 1.
- 2The third entry must agree: 2·2 + 1 = 5, which holds, so the answer is yes.
Key terms
- Vector Space
- A vector space is a set closed under addition and scalar multiplication that satisfies the standard algebraic rules of Rⁿ.
- Subspace
- A subspace is a subset of a vector space that is itself a vector space under the same operations.
- Linear Combination
- A linear combination is a sum of vectors each multiplied by a scalar weight.
- Linearly Dependent Set
- A linearly dependent set has a combination with weights not all zero that adds to the zero vector.
Subspaces, Spanning Sets and Linear Independence FAQ
Is the set of solutions of Ax = b a subspace?
Only when b is the zero vector. With a nonzero right-hand side the zero vector is not a solution, so the set fails the Subspace Test, although it is still a shifted copy of the null space.
Which vectors do I keep after finding a dependence?
Keep the original vectors whose columns contain pivots in the reduced matrix. They are independent and have exactly the same span as the full set, so nothing is lost.
Does a spanning set have to be linearly independent?
No. A spanning set can contain redundant vectors. Removing the vectors without pivots in the reduced matrix leaves an independent set with exactly the same span, which is then a basis.
Exam move
Keep the two setups side by side in your notes: target vector after the bar for span, zeros after the bar for independence. Most lost points on these questions come from mixing them up. Write the one-line setup for each question type on a card and quiz yourself on which applies before reducing anything.
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