Midterm 2 Review

Sum of vectors, scalar multiples, linear combinations

• The sum of two vectors is found by adding their corresponding components.
  If $\vec{u}=(u_1,u_2,...,u_n)$ and $\vec{v}=(v_1,v_2,...,v_n)$, then:

  $$\vec{u}+\vec{v}=(u_1+v_1,u_2+v_2,...,u_n+v_n)$$

• A scalar multiple of a vector means multiplying each component of the vector by a constant:

  $$c\vec{v}=(c{v}_1,c{v}_2,...,c{v}_n)$$

• A linear combination of vectors ${\vec{v}}_1,{\vec{v}}_2,...,{\vec{v}}_n$ is any expression of the form:

  $$c_1{\vec{v}}_1+c_2{\vec{v}}_2+\cdots +c_n{\vec{v}}_n$$

  where $c_1,...,c_n$ are scalars. You’re combining vectors using addition and scalar multiplication.

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Write a vector as a linear combination of other vectors

• Given a set of vectors and a target vector, determine if the target vector can be written as a linear combination of the set.
• Set up a matrix equation: $$A\vec{x}=\vec{b}$$ where $A$ is a matrix with the given vectors as columns, and solve for $\vec{x}$.
• If a solution exists, then $\vec{b}$ is a linear combination of the columns of $A$.

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Describe the zero vector in various vector spaces

• ${\mathbb{R}}^n$: The vector $(0,0,...,0)$ with all components zero.
• $P_n$: The zero polynomial: $0x^n+0x^{n-1}+\cdots +0$.
• $M_{mn}$: The $m\times n$ matrix where all entries are 0.

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Determine if a set is a vector space

To be a vector space, a set must:

• Contain the zero vector.
• Be closed under addition and scalar multiplication.
• Satisfy 10 axioms (like associativity, identity, inverse, distributivity, etc.).
• Usually, we test whether a subset of a known vector space satisfies the vector space axioms.

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Determine if a set is a subspace

A subspace of a vector space must:

1. Contain the zero vector.
2. Be closed under vector addition: If $\vec{u},\vec{v}$ are in the set, then $\vec{u}+\vec{v}$ is too.
3. Be closed under scalar multiplication: If $\vec{v}$ is in the set and $c$ is any scalar, then $c\vec{v}$ is in the set.

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Determine if a set spans ${\mathbb{R}}^n$

• A set of vectors spans ${\mathbb{R}}^n$ if any vector in ${\mathbb{R}}^n$ can be written as a linear combination of them.
• Equivalent to: the matrix formed by the vectors has a pivot in every row after row reduction.

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Determine if vectors are linearly dependent or independent

• A set is linearly dependent if:

  $$c_1{\vec{v}}_1+\cdots +c_n{\vec{v}}_n=\vec{0}$$

  has a non-trivial solution (not all $c_i=0$).

• A set is linearly independent if the only solution is the trivial one: $c_1=c_2=\cdots =c_n=0$.

• To test: Form a matrix with the vectors as columns and solve $A\vec{x}=\vec{0}$.

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Determine if a set is a basis

A set of vectors is a basis if:

• They span the vector space.
• They are linearly independent.

If the dimension of the space is $n$, you need exactly $n$ linearly independent vectors to form a basis.

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Find the rank and nullity of a matrix

• Rank: The number of pivot columns in the row-reduced form of the matrix. Represents the dimension of the column space.
• Nullity: The number of free variables in the solution to $A\vec{x}=0$.
• Formula: $$\text{rank}(A)+\text{nullity}(A)=\text{number of columns}$$

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Basis for a subspace spanned by a set

• To find a basis for the subspace spanned by a set:
  • Put the vectors as rows or columns of a matrix.
  • Row reduce.
  • The original vectors corresponding to the pivot columns form the basis.

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Basis and dimension of solution space to $A\vec{x}=0$

• Solve the homogeneous system.
• Express the solution in parametric vector form.
• The vectors that multiply the free variables form a basis.
• The number of these vectors = dimension of the null space.

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Determine if a vector is in the column space

• Given matrix $A$ and vector $\vec{b}$:
  • Form the augmented matrix $[A\mid \vec{b}]$.
  • If the system is consistent (no contradictions), then $\vec{b}$ is in the column space of $A$.

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Transition matrix from $B$ to $B^{\prime }$, or $B^{\prime }$ to $B$

• A transition matrix changes coordinates from one basis to another.
• To find matrix from $B$ to $B^{\prime }$:
  • Express each vector of $B^{\prime }$ as a linear combination of vectors in $B$.
  • Place these coordinates as columns of the matrix.

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Wronskian of a set of functions

• Given functions $f_1,f_2,\dots ,f_n$, their Wronskian is: $$W(f_1,\dots ,f_n)=\left|\begin{array}{ccc}{f}_1& f_2& \dots \\ f_1^{\prime }& f_2^{\prime }& \dots \\ ⋮& ⋮& \ddots \end{array}\right|$$
• If $W\ne 0$ on an interval, the functions are linearly independent on that interval.

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Write a proof based on dependence, independence, subsets

• Use definitions:
  • If one vector is a linear combination of others, the set is dependent.
  • A subset of an independent set is also independent.
  • Adding a vector to a dependent set keeps it dependent.
• Structure your proof by assuming or proving dependence/independence based on these principles.

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Vector operations and notation

• Length (magnitude):

  $$\Vert \vec{v}\Vert =\sqrt{v_1^2+v_2^2+\cdots +v_n^2}$$

• Distance between vectors:

  $$\text{dist}(\vec{v},\vec{w})=\Vert \vec{v}-\vec{w}\Vert$$

• Inner product (dot product):

  $$\vec{v}\cdot \vec{w}=v_1{w}_1+v_2{w}_2+\cdots +v_n{w}_n$$

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Angle between two vectors

• Formula: $$\cos (\theta )=\frac{\vec{v}\cdot \vec{w}}{\Vert \vec{v}\Vert \Vert \vec{w}\Vert }$$
• Use this to find the angle or to verify orthogonality ($\cos (\theta )=0$ when vectors are perpendicular).

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Orthogonal and orthonormal sets

• Orthogonal: All pairs of vectors have dot product 0.
• Orthonormal: Orthogonal set where each vector has length 1.
• Check:
  • ${\vec{v}}_i\cdot {\vec{v}}_j=0$ for $i\ne j$
  • $\Vert {\vec{v}}_i\Vert =1$ for all $i$

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Unit vector in same direction

• A unit vector has length 1.
• To create a unit vector in the same direction as $\vec{v}$: $$\hat{v}=\frac{\vec{v}}{\Vert \vec{v}\Vert }$$

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Gram-Schmidt Process

• Converts a set of linearly independent vectors into an orthogonal (or orthonormal) set.

Given ${\vec{v}}_1,{\vec{v}}_2$:

1. Let ${\vec{w}}_1={\vec{v}}_1$

2. Subtract the projection of ${\vec{v}}_2$ onto ${\vec{w}}_1$:

   $${\vec{w}}_2={\vec{v}}_2-\frac{⟨{\vec{v}}_2,{\vec{w}}_1⟩}{⟨{\vec{w}}_1,{\vec{w}}_1⟩}{\vec{w}}_1$$

3. Normalize to get orthonormal vectors:

   $${\hat{w}}_i=\frac{{\vec{w}}_i}{\Vert {\vec{w}}_i\Vert }$$

Remember to normalize at the end if the problem asks for orthonormal vectors.

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linear math