Discussion Overview
The discussion revolves around determining the value(s) of \( h \) for which the vector \( b \) lies in the plane spanned by the vectors \( a_1 \) and \( a_2 \). The context includes mathematical reasoning and linear algebra concepts related to linear combinations and augmented matrices.
Discussion Character
Main Points Raised
- One participant poses the question of what value(s) of \( h \) would place \( b \) in the plane spanned by \( a_1 \) and \( a_2 \), suggesting that \( b \) must be a linear combination of \( a_1 \) and \( a_2 \).
- A hint is provided that \( b \) can be expressed as \( b = v a_1 + w a_2 \) for some constants \( v \) and \( w \).
- Another participant constructs the augmented matrix corresponding to the linear combination and performs row reduction to find \( h \), arriving at the conclusion that \( h = 3 \) along with values for \( v \) and \( w \).
- A correction is made regarding the signs of \( v \) and \( w \), suggesting that they should be \( -7 \) and \( -2 \), respectively, while still agreeing on \( h = 3 \).
- One participant notes that the original question only asked for \( h \), implying that the additional information about \( v \) and \( w \) may not have been necessary.
Areas of Agreement / Disagreement
Participants generally agree that \( h = 3 \) is the value that allows \( b \) to be in the plane spanned by \( a_1 \) and \( a_2 \). However, there is a disagreement regarding the signs of \( v \) and \( w \>, with one participant suggesting they are \( -7 \) and \( -2 \).
Contextual Notes
The discussion involves assumptions related to linear combinations and the properties of augmented matrices, but these assumptions are not explicitly stated or resolved.