Linear Algebra
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7 Terms
Subspace of R^n
A subspace of W in R^n is a non-empty subset that is closed under vector addition and scalar multiplication. To prove this:
1. Zero Vector: The zero vector 0 is in W.
2. Closure Under Addition: For any vectors u,
𝐯∈𝑊, the sum of u+v is is also in W.
3. Closure Under Scalar Multiplication: For any vector 𝐮∈𝑊 and any scalar 𝑐∈R, the product cU is also in W.
Automatic no's:
1. No Zero Vector (Not Equal to Zero)
2. Inequalities
3. NonLinear Equations (Absolute Values, Exponents, Trig)
4. Absolute Values
Understand why the span of any set of vectors is a subspace.
It is closed under the two fundamental vector operations: addition and scalar multiplication, and it contains the zero vector.
Column Space (Col𝐴)
Row Space (Row𝐴)
Null Space (Nul𝐴)
Helpful Tip for Column, Row, Null
Basis of a Subspace
A basis of a subspace 𝑆 is a set of vectors that acts as a coordinate system, satisfying two conditions: they span the subspace (can form all vectors in 𝑆 via linear combinations) and are linearly independent. It is the minimal spanning set, where every vector in the subspace is represented uniquely.
