MATH1012 - Lecture 5

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MATH1012 - Lecture 4
Skeleton Slides

Proving a Spanning Space

• Given subspace $S$, how do we prove $T$ is its spanning space?
  • Must show that every linear combination of the vectors are in $S$
  • Must show every part of $S$ can be created using vectors in $T$
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Find a Spanning Set

To find a spanning space of a subspace, choose vectors from within the subspace until you can recreate the set with only that set of vectors. You will need a vector that can influence each dimension. Thus, for a set with an $x,y$ and $z$ component, your spanning set must have a vector with an $x,y$ and $z$ component!

Because a subspace is closed under addition and multiplication, you can choose any vector without the worry of your spanning space going beyond the subspace. When choosing a vector, don’t choose one that you can already make with the vectors you have already chosen. Try and use the least amount of vectors possible!

Linear Dependence

• A set of vectors is linearly dependent if one of them is a linear combination of the others
• A set of vectors is linearly independent if it is not linearly dependent

Linear Independence Test

In ${\mathbb{R}}^n$, as set $T=\{v_1,v_2,...v_k\}$ is linearly independent if and only if the homogeneous system of $n$ linear equations: ${\alpha }_1{v}_1+{\alpha }_2{v}_2+\cdot \cdot \cdot +{\alpha }_k{v}_k=0$
in the unknowns ${\alpha }_1,{\alpha }_2,...,{\alpha }_k$ has the unique solution: ${\alpha }_1={\alpha }_2=...={\alpha }_k=0$

• This can be tested through using Gaussian Elimination

Properties of Dependence

• A subset of a linearly independent set is linearly independent
• A superset of a linearly dependent set is dependent

Having a dependent set means there is a redundancy

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MATH1012 - Lecture 6
Annotated Slides