Linear combination and vector equations in RnR^n - Linear Equations with Matrices

Linear combination and vector equations in RnR^n

Lessons

Notes:
A matrix with one column is called a column vector. They can be added or subtracted with other column vectors as long as they have the same amount of rows.

Parallelogram Rule for Addition: if you have two vectors uu and vv, then u+vu+v would be the fourth vertex of a parallelogram whose other vertices are u,(0,0)u,(0,0),and vv

Here are the following algebraic properties of Rn\Bbb{R}^n
1. u+v=v+uu+v=v+u
2. (u+v)+w=u+(v+w)(u+v)+w=u+(v+w)
3. u+0=0+u=uu+0=0+u=u
4. u+(u)=u+u=0u+(-u)=-u+u=0
5. c(u+v)=cu+cvc(u+v)=cu+cv
6. (c+d)u=cu+du(c+d)u=cu+du
7. c(du)=(cd)(u)c(du)=(cd)(u)
8. 1u=u1u=u

Given vectors v1,,vpv_1,\cdots,v_p in Rn\Bbb{R}^n with scalars c1,,cpc_1,\cdots,c_p, the vector xx is defined by

x=v1c1++vpcpx=v_1 c_1+\cdots+v_p c_p

Where xx is a linear combination of v1,,vpv_1,\cdots,v_p.

The linear combinations of v1,,vpv_1,\cdots,v_p is the same as saying Span{v1,,vpv_1,\cdots,v_p}.
  • Intro Lesson
    Vector Equations in Rn\Bbb{R}^n Overview:
  • 1.
    Calculating Vectors in Rn\Bbb{R}^n
    Consider the two vectors Calculating vectors in R^n, vector 1, and Calculating vectors in R^n, vector 2. Compute:
  • 4.
    Linear Combinations with Known terms
    Determine if bb is a linear combination of a1a_1, a2a_2 in part a. Determine if bb is a linear combination of a1a_1, a2a_2, and a3a_3 in part b and c.
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Linear combination and vector equations in RnR^n

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