Status: Tags: #cards/math232/unit1 Links: Vectors
Vector Subspace
Principles
- Every subspace contains the zero vector Subspace ;; a subset of a space, closed under addition and scalar multiplication.
Trivial subpsaces ;; subspaces consisting of either 0 or R^n
Types of subspaces of $R^2$ are ;; the trivial subspace, lines through origin, and $R^2$
Types of subspaces of $R^3$ are?
- trivial subspace
- lines through the origin
- planes through the origin
- all of $R^3$
Span are subspaces
Types
Determining Subspace
- It is a subspace if it is a line passing through origin
- Reduce to two variables hehe
Determining dimension of subspace ?
- Find number of vectors in basis
Bases of Subspaces
Bases ;; set of vectors in a subspace that spans a subspace AND in which all of the vectors are linearly independent is a basis for the subspace
Proving subspace of Vector Space
?
- Check if zero vector works
- Check if closed under addition
- Take any vector x and y such that x1,x2,x3 (- R and any y1,y2,y3 (- R
- Add two vectors together
- Demonstrate “such that rules” under vector addition
- Check scalar multiplication
- Take any vector and scalar such that x1,x2,x3 (- R and any scalar c (- R
- Demonstrate vector under scalar multiplication
- Demonstrate “such that rules” under scarlar multiplication
Prove sx + ty (- S
- Check if all rules of a Vector Space work on S
Examples
Consider all vectors in R^4 with the peoperty that $x_1$ + $2x_2$ = 0. Is this set a subspace of $R^4$? ?
- Use the equation to substitute any components
- Check to see if components of new vector are in R
- Check 0 vector
- Check if closed under linear combination by creating 2 general vectors x and y
- Combine, see if they match sx + ty (- S
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References:
Created:: 2022-01-17 15:12