Linear combination and vector equations $R^n$

Lessons

• Introduction
  Vector Equations in $\Bbb{R}^n$ Overview:
  a)

  Vectors in $\Bbb{R}^2$
  • Column vectors with 2 rows
  • Adding, subtracting, and multiplying 2D vectors
  • Graphing vectors in 2D
  • Parallelogram Rule for Addition
  
  
  b)

  Vectors in $\Bbb{R}^3$
  • Column vectors with 3 rows
  • Adding, subtracting, and multiplying 3D vectors
  • Graphing vectors in 3D
  
  
  
  c)

  Vectors in $\Bbb{R}^n$
  • Column vector with $n$ rows
  • Algebraic properties
  
  
  d)

  Linear Combinations and Spans
  • Vectors and weights
  • Vector equations
  • Finding a linear combination with row reduction
  
  

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• 1.
  Calculating Vectors in $\Bbb{R}^n$
  Consider the two vectors , and . Compute:
  a)

  $u+2v$
  
  
  b)

  $2u-v$
  
  
  c)

  $5u+0v$
  
  

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• 2.
  Converting Systems Of Equations And Vector Equations
  Write the given systems of equations as a vector equation.
  
  $2x_1+x_2-5x_3=4$ $x_1+3x_2+2x_3=1$ $-4x_1-x_2-8x_3=-2$

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• 3.
  Write the given vector equation has a system of equations
  

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• 4.
  Linear Combinations with Known terms
  Determine if $b$ is a linear combination of $a_1$, $a_2$ in part a. Determine if $b$ is a linear combination of $a_1$, $a_2$, and $a_3$ in part b and c.
  a)

  
  
  
  b)

  
  
  
  c)

  
  
  

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• 5.
  Linear Combinations with Unknown terms
  For what value(s) of $k$ is $b$ in the plane spanned by $a_1$ and $a_2$ if:
  

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